...as per title, I'm wondering what the effect of striking the prohibition (removing it) on spells being cast on the same turn as a spell that takes a bonus action would be.

Sorcerers would become overpowered

Most spells that take a bonus action to cast are relatively minor, like Misty Step or Healing Word. They are often utility spells, or spells that enhance a physical attack (like a paladin's Smite spells). Most of them wouldn't unbalance play if they were permitted during the same round as another leveled spell.

But whether or not unbalancing bonus action spells exist by default, a sorcerer's Quickened Spell metamagic is highly dependent on the current rules on bonus action spells. Sorcerers can cast Action spells using bonus actions for the cost of 2 sorcery points, which is far more powerful if they could then cast another leveled spell again with their action. This is particularly problematic if you consider the Sorcerer/Warlock multiclass, which can use Warlock spell slots to regain Sorcery points as a short rest resource.

The ability to cast two fireball spells in a single turn multiple times per combat at low levels is game changingly problematic.

Spellcasters with misty step would become even more slippery

If a spellcaster ends up in melee range, misty step is a cheap, guaranteed option to get out without getting slapped by an attack of opportunity. It's basically a dash and disengage in one bonus action (and more, depending on what the teleportation lets you bypass). The only downsides are using a bonus action (if you even have any use for it), the spell slot (2nd level, pretty cheap), and the bonus action restriction.

With the restriction, a spellcaster has many options for what to do about an attacker in melee range, but they all have consequences. Disable/push them with a spell? Might come at a cost of using another spell on another target, and could fail if they make their save. Using misty step is a great option, but against may come at the cost of using an important spell. With the way it is, even a melee attacker that hasn't gotten a chance to attack the spellcaster yet (they dashed up or something) can have a significant impact.

Without the restriction, you could try to disable/push them with a spell, fail, and then misty step away once you see it didn't work. Or you could go ahead and cast your more important spell on your more important target and still avoid taking any damage from the melee attacker.

It basically adds this layer of safety that changes the counterplay between spell casters and melee attackers, and reduces the impact of a melee attacker getting into range of a spellcaster.