The whole idea that Schummer had to back the cloture vote is flawed. So, the Republicans would blame the Democrats for shutting down the government. And so? The Democrats could just as easily frame the Republicans as the ones closing down the government.
The whole idea that they would be blamed for SS checks not arriving is easily countered. The Republicans had/have multiple avenues toward keeping the government functioning:
* Pass an actual budget
* Amend the continuing resolution to make it more palatable for Democrats and their voters
* Nuke the filibuster and pass the CR with their own votes
What the Dems have now done is to remove any reason for anyone to ever vote for them again. They are unwilling to do anything to stop the destruction of democracy in the US. They are unwilling to do anything to advance the concerns of the people who voted for them. All they seem to care about is to look like the adults in the room. Well, they've tried that for 25 years now to no avail. Maybe it is time to choose a different strategy, cuz this one ain't working.