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Universal health insurance which is affordable for everyone can only work when the healthy people subsidize the people with chronic health problems.

If you have a completely unregulated private insurance sector, then the private insurers would be free to choose their customers. Given that chance they would reject anyone with pre-existing conditions, because these people would become a net cost instead of a net benefit for the insurance company. So the private insurers would grab all the "good" customers while the government provided health insurance is stuck with the "bad" ones. That means the cost for the taxpayer to provide that health insurance would increase drastically (either because the state needs to subsidize the health insurance system or because the premiums must be increased) while the private insurers make a hefty profit.

There are basically two solutions for this problem.

Either you strongly regulate the private insurers in order to prevent them from cherry-picking their customers. You might make it illegal for insurance companies to reject customers or to take different prices for the same plan from different customers. But it's hard to outlaw things like targeted advertisement or insurance plans heavily optimized for the preferences of certain target groups.

Or you make the government health insurance mandatory for all citizens. This effectively removes the market for private health insurance and only leaves a market for "luxury" health services which go beyond the regular health care.

Universal healthcare systems can be harmed by competition from private healthcare, such as when the private companies pay more and make it difficult for the state system to recruit staff. It can also make it harder for national systems to buy medicine at reasonable prices, when manufacturers can rely on the private sector paying more and people desperate for it.

There are also issues of fairness, as when private treatment is available those who can afford it can bypass queues in the state system by getting specific procedures done privately, and then return to the state system when it suits them.

Another issue is the state picking up the bill for failures in the private sector, such as a recent incident in the UK where sub-standard breast implants fitted by a company that subsequently went bust had to be fixed at the taxpayer's expense.

One reason, which is not touched upon by other answers, is that private insurance coverage would increase health care costs for everyone, including those universally covered.

One of the major points of savings in a single-payer system is the reduced administrative overhead. People automatically think of the reduced overall overhead and elimination of redundancies for a single payer vs multiple insurance companies, but what has as much impact is the reduction in administrative staff and systems needed for health care providers to handle billing.

If you have a single payer, there are only one set of reimbursements, one method of coding and bundling procedure coding for insurance claims, one set of rules and prerequisites for allowable claims. For every insurer you add, you are adding the complexity of tracking a parallel system for the differences between insurers, and then the need to track which patients align with which sets of insurance rules, so the provider and the patient don't unexpectedly get stuck with a service that can't be paid for. This administrative overhead drives up the cost of all health care for that provider, and that is reflected in the billing rates the provider will charge to all patients.

Clinics or healthcare systems have an army of administrators and analysts just to manage the billing of services. Smaller providers have to pay for services to manage it for them. That's huge overhead and no value added.

So, if you have a system that is paid for by tax dollars, allowing private insurers, to a certain degree, would require all taxpayers taking on an added burden for the benefit of that private insurer, and not just the premium paying customers of that insurance company.

EDIT: Interestingly enough, it appears that there are some people aware of the high administrative cost of healthcare and insurance in the USA, which is why we're supposed to source everything.

For those challenging the premise of administrative inefficiency specifically from billing complexity -

METHODS
For the United States and Canada, we calculated the administrative costs of health insurers, employers' health benefit programs, hospitals, practitioners' offices, nursing homes, and home care agencies in 1999. We analyzed published data, surveys of physicians, employment data, and detailed cost reports filed by hospitals, nursing homes, and home care agencies. In calculating the administrative share of health care spending, we excluded retail pharmacy sales and a few other categories for which data on administrative costs were unavailable. We used census surveys to explore trends over time in administrative employment in health care settings. Costs are reported in U.S. dollars.

RESULTS
In 1999, health administration costs totaled at least $294.3 billion in the United States, or $1,059 per capita, as compared with $307 per capita in Canada. After exclusions, administration accounted for 31.0 percent of health care expenditures in the United States and 16.7 percent of health care expenditures in Canada. Canada's national health insurance program had overhead of 1.3 percent; the overhead among Canada's private insurers was higher than that in the United States (13.2 percent vs. 11.7 percent). Providers' administrative costs were far lower in Canada.

Between 1969 and 1999, the share of the U.S. health care labor force accounted for by administrative workers grew from 18.2 percent to 27.3 percent. In Canada, it grew from 16.0 percent in 1971 to 19.1 percent in 1996. (Both nations' figures exclude insurance-industry personnel.)

CONCLUSIONS
The gap between U.S. and Canadian spending on health care administration has grown to $752 per capita. A large sum might be saved in the United States if administrative costs could be trimmed by implementing a Canadian-style health care system.

New England Journal of Medicine: Costs of Health Care Administration in the United States and Canada

ABSTRACT
Billing and insurance–related functions have been reported to consume 14 percent of medical group revenue, but little is known about the costs associated with performing specific activities. We conducted semistructured interviews, observed work flows, analyzed department budgets, and surveyed clinicians to evaluate these activities at a large multispecialty medical group. We identified 0.67 nonclinical full-time-equivalent (FTE) staff working on billing and insurance functions per FTE physician. In addition, clinicians spent more than thirty-five minutes per day performing these tasks. The cost to medical groups, including clinicians’ time, was at least $85,276 per FTE physician (10 percent of revenue).

Health Affairs - Peering Into The Black Box: Billing And Insurance Activities In A Medical Group (Sakowski, Kah, Kronick, Newman, Luft)

You can see that there is significant billing and insurance costs, and that those costs for providers is much lower in the Canadian system where private insurance is an addition to their system, not the primary focus.

Both of those studies were cited by the following NY Times article about the high cost of healthcare administration in the USA.

A widely cited study published in The New England Journal of Medicine used data from 1999 to estimate that about 30 percent of American health care expenditures were the result of administration, about twice what it is in Canada. If the figures hold today, they mean that out of the average of about $19,000 that U.S. workers and their employers pay for family coverage each year, $5,700 goes toward administrative costs.

Such costs aren’t all bad. Some are tied up in things we may want, such as creating a quality improvement program. Others are for things we may dislike — for example, figuring out which of our claims to accept or reject or sending us bills. Others are just necessary, like processing payments; hiring and managing doctors and other employees; or maintaining information systems.

The portion more specifically about billing costs -

Hospitals are not the only source of high administrative spending in the United States. Physician practices also devote a large proportion of revenue to administration. By one estimate, for every 10 physicians providing care, almost seven additional people are engaged in billing-related activities.

It is no surprise then that a majority of American doctors say that generating bills and collecting payments is a major problem. Canadian practices spend only 27 percent of what U.S. ones do on dealing with payers like Medicare or private insurers.

Another study in Health Affairs surveyed physicians and physician practice administrators about billing tasks. It found that doctors spend about three hours per week dealing with billing-related matters. For each doctor, a further 19 hours per week are spent by medical support workers. And 36 hours per week of administrators’ time is consumed in this way. Added together, this time costs an additional $68,000 per year per physician (in 2006). Because these are administrative costs, that’s above and beyond the cost associated with direct provision of medical care.

In JAMA, scholars from Harvard and Duke examined the billing-related costs in an academic medical center. Their study essentially followed bills through the system to see how much time different types of medical workers spent in generating and processing them.

At the low end, such activities accounted for only 3 percent of revenue for surgical procedures, perhaps because surgery is itself so expensive. At the high end, 25 percent of emergency department visit revenue went toward billing costs. Primary care visits were in the middle, with billing functions accounting for 15 percent of revenue, or about $100,000 per year per primary care provider.

“The extraordinary costs we see are not because of administrative slack or because health care leaders don’t try to economize,” said Kevin Schulman, a co-author of the study and a professor of medicine at Duke. “The high administrative costs are functions of the system’s complexity.”

NY Times: The Astonishingly High Administrative Costs of U.S. Health Care (July 16, 2018)

If you do not have the laws in place to deal with private health insurance, the simple approach is to ban it altogether.

Private health insurance can do plenty of bad things unless you have the laws to prevent them. To name just a few:

1. Only accept young, healthy, male, non smoking, and generally low risk clients. This increases the average costs of the remaining population, all of whom happen to be on public health insurance.


2. Reject renewal for sick clients. This pushes sick people to public health insurance, increasing cost of public health care.


3. Refuse to pay for certain expensive treatments. This pushes people requiring expensive treatments to public health care, increasing cost of public health insurance.


4. Bad faith advertising.

And there's much more than just these. All of these exploits have to be addressed with laws and control mechanisms, which need to be negotiated between various lawmakers and interest groups first.

I think the law doesn't regulate (or only indirectly regulates) the insurance providers.

The main point is that it regulates the service providers (i.e. the health care providers, e.g. doctors).

I think the law is that you (a health care provider) are legally not allowed to charge privately (e.g. to charge a patient or a private insurance company) for any type of service (e.g. surgery) that would be paid for by the government's universal health insurance (which includes any or every kind of mainstream health care).

So it's not especially that there's no private health insurance, it's that there's no private health care (people choose their own doctors, but the doctors and hospitals are paid by the government).

And the rationale for that restriction (i.e. no private health care) is that they (the government i.e. the people) don't want a "two-tier" health system, where rich people (and/or people with private insurance) get better health care, and other people get worse health care.

I think a Canadian Health Minister (i.e. a politician) once said that in the States anyone can get however much health care each person can afford, whereas in Canada everyone can get however much they need.

I think I heard that leads to cases where e.g. a professional athlete goes to the States to get a soft-tissue injury looked at quickly (and at private cost), but generally I don't think that (i.e. cross-border shopping) is a big factor.

Private insurance in Canada can be obtained for things whose whose cost is not covered by the universal (provincial) plans -- including e.g. prescriptions and prostheses (the universal plan pays for medical services), dental care, eye glasses, maybe physical rehabilitation, pay your lost wages, travel insurance, cosmetic surgery, that kind of thing, maybe a private room in a hospital -- that insurance is often a (taxable) "benefit" paid by your employer. But there's no insurance to pay for surgery or to visit a family doctor, because those costs and services are covered by the government plans.

Or these "supplementary" costs listed above (e.g. prescriptions) which aren't covered universally may be covered by "welfare" payments, e.g. based on means-tests for people with no or low incomes, or people who have catastrophically high prescription drug costs.

A further result may be (I don't know) that the amount a service provider (e.g. a doctor) can charge is lower, too, that it might be otherwise -- since the government has a monopoly on paying them.

I think I read that the costs of prescription drugs, too, is lower in the Canada than in the States, even though people pay for those out of their own pockets. I'm not sure why that's so, I think it's that the government can (and must) to permit whatever drugs are sold and that it negotiates a price with the drug companies, and that a new drug which is too expensive isn't permitted -- but that's a whole other topic.

Wikipedia (however I'm not sure that this tells the whole story):

Six of Canada's ten provinces used to ban private insurance for publicly insured services to inhibit queue jumping and so preserve fairness in the health care system. In 2005, the Supreme Court of Canada ruled that in Quebec, such bans are unconstitutional if the waiting period for care is excessively long. However, this ruling only applies within the Province of Quebec. A second court challenge is currently underway to determine whether the prohibition of private parallel health care violates the patients' right to life, liberty, and security under Section 7 of the Canadian Charter of Rights and Freedoms.

The criticisms section is long -- but perhaps that's failing to talk about the good news; or the good news is easily summarised:

Health cards are issued by provincial health ministries to individuals who enroll for the program in the province and everyone receives the same level of care. There is no need for a variety of plans because virtually all essential basic care is covered, including maternity

I don't think the service is much worse than elsewhere, and think that it's more-or-less comparable to other developed countries -- but is perhaps unusual (compared to e.g. the UK or France) in not even permitting private health care.

Based on the article The illegality of private health care in Canada written in 2001, which was added to the OP, perhaps I should revise my answer:

• What I said is true of some but not all provinces


• Details vary, e.g. in some provinces (e.g. Ontario) private health care is permitted but a physician isn't allowed to charge more than they'd be paid in the public system


• In other provinces a physician may charge more but if they do then they're not paid by the government at all (i.e. the user pays the whole cost not just the difference), and private insurance isn't permitted.

It says ...

We conclude by noting that in Canada, the absence of a private system is not due to the illegality of private health care per se. Private insurance for the kinds of medically necessary hospital and physician services that the public service is meant to cover is illegal in only 6 provinces.

... but I think these 6 are all the biggest ones (i.e. the remaining 4 are New Brunswick, Newfoundland, Nova Scotia and Saskatchewan -- or approximately all the biggest ones, because Saskatchewan is more populous than Prince Edward Island).

It also says ...

Arguably, Canada already has a 2-tier health care system because of the rigid division between medically necessary hospital and physician services (enshrined and protected in the Canada Health Act) and other kinds of goods or services for which there is significant private financing, such as drugs and home care.

... which I'm not sure is true. I'm sure it's true that e.g. dental care is difficult (barely available) for poor people, and home care too, and access to health care is sparce in small, remote communities -- but maybe drugs are available, either because of welfare or e.g. the Trillium drug plan for people whose drug costs more than 4% of income.

Anyway you might read the article if you want all its details and conclusion. It claims at the start that ...

Our survey revealed multiple layers of regulation that seem to have as their primary objective preventing the public sector from subsidizing the private sector, as opposed to rendering privately funded practice illegal.

... but I'm not sure that's so -- I think that "preventing subsidies" might be a "means or mechanism", rather than an "objective". Instead I think the objective might be as defined in the Canada Health Act i.e.

Universality: All insured residents are entitled to the same level of health care

The authors have several other articles on this and related topics for example here.