I’m sure there is something that I am missing (certain vote percentages, loopholes, who knows), but to me the logic seems that if Republicans had the ability to pass the funding measure and then didn’t, wouldn’t the shutdown be the Republicans fault?

I think there are a couple things: first, the new Democratic majority; and second, the filibuster/cloture process in the Senate.

Yesterday (3 Jan.) was the first day of the 116th Congress. As of yesterday, the Democrats have a majority in the House (235 to 199, one disputed seat) and the Republicans have a majority in the Senate (53 to 47). At this point and going forward, both parties bear responsibility for passing or failing to pass spending bills.

However, before yesterday, the Republicans had a 236 to 196 majority in the House (three vacant seats) and a 51 to 49 majority in the Senate. In December, the (Republican) House passed a spending bill with funding for the president's proposed border wall. The Senate unanimously passed an alternative short-term spending measure without border wall funding, which the president then threatened to veto. Following that, the Senate Majority Leader stated that he would not support (or presumably schedule a vote for) any bill that the president threatened to veto.

Also, almost all bills in the Senate require 60 senators to invoke "cloture" in order to end debate and vote. Bills that fail to receive cloture are "filibustered," and given that Senate Republicans had an extremely slim majority in the last Congress, invoking cloture against a united Democratic conference was quite tough. Even in the new Congress, invoking cloture will be tricky for polarizing legislation (e.g. anything dealing with "the wall"), albeit marginally easier for the Republicans than in the last Congress.

In my opinion, anyone who assigns blame or responsibility for the shutdown to one party exclusively is trying to spin the facts to fit a partisan or ideological narrative. How you assign blame depends on your personal beliefs, what you think about the majorities in Congress and what you think about the filibuster.

This isn't really that difficult: It's a typical standoff in which neither side wants to budge. Democrats could vote for the wall, but haven't. So, sure, it can be considered at least partially their fault. That's not to say Trump isn't to blame, either. But, if ending the shutdown were enough of a priority, then Democrats could simply vote for the money for the wall, and be done with it. If it's not, the standoff will continue.

Passing a spending bill in the Senate requires bi-partisan cooperation because it requires overcoming a 60-vote requirement. The President demanded that any spending bill include a line for $5 billion (about 0.13% of the total yearly spending) for a border barrier (which he colloquially called a "wall").

The Republicans were willing to include the line for this spending. The Democrats were not. By this logic, the Democrats have a partial responsibility for the bill not passing.

Clearly they are not the only ones to blame. The President's unwillingness to give up on funding of "the wall" is to blame as well. But using the cloture rules to hold up a spending bill over a 0.13% spending line is the reason (or logic) for why the President is putting the failure to pass the spending bill at the Democrats' feet.

Edit: in light of the answer to this question, this answer needs clarification.

Assuming that bills do expire when a Congress ends, the bill passed by the Republican House expired when the new Congress was sworn in. So while cloture rules were the reason why the spending bill was never passed by the outgoing Congress, it is not the reason why the new Congress, sworn-in in January of 2019, did not pass a spending bill.

In addition to having enough votes to force a filibuster in the Senate, as of January of 2019, the Democrats also have control of the House of Representatives. This gives them more power to stop any spending items that the President demanded.

If Republicans have a majority in the House and the Senate, and also have control over the executive branch, how could it be possible that the Democrats are responsible for the shutdown?

Answer: The Republicans' majority in the Senate isn't large enough.

In order to end debating a bill, 60 senators have to agree to start voting. Because the Republicans only held either 51 (just before the 2018 election) or 53 (after the 2018 election), the Democrats have enough votes to continue debating certain bills forever. This is known as 'filibusting' a bill.

As a result, the Republicans were not able to pass a bill that included the requested money for a border wall, even when they had majorities in both chambers of Congress.

The simple answer is that they say so - they do not necessarily have to give a credible reason for why it is the Democratic Party's fault for people to believe them. For some, the President's words are enough.

For the more discerning mind, I'll take a bit of @Alexander O'Mara 's answer to explain. To summarize - in December, the Senate passed a bipartisan budget that did not include funding for the house. Right-leaning news sources criticized this, prior to the President's signage, as the president folding on the border wall. Trump refused to sign this bill, and over the Winter recess, the government shutdown began.

Now in January, the Democrats are submitting a very similar budget proposal to the one they submitted in December - and the argument being made by Mitch McConnell for not bringing it to the House floor is simple - the bill has already failed to get Presidential approval, and there is not enough support from his own party to override the Presidential veto, so the gesture from the Democrats is symbolic at best, and pointless at worst.

This echoes President Trump's views - that any negotiations with standing Senate leaders in the Democratic Party that do not address Trump's desire for border wall funding are pointless - because they will not meet his support and, likewise, not meet Republican party support in the House.

To summarize, while the bill sent to the House in December had enough support to make it to the President's desk, the lack of border wall funding in that bill had the President veto it. The Senate is now trying to push the old bill - and the Republican-Controlled house is rejecting it, with the explanation that it already does not meet with the President's approval.