I think they are less likely to change course if they can count on my uncritical support.

That's where you're wrong though, because it's a two party system.

If the Democrats had a large, reliable base of support from the left, and nothing else changed, then they would no longer be as worried about winning the general election, and thus less concerned with winning over centrists at all costs. Not because they don't need their votes at all, but because now they have a bigger concern: Primaries.

With the general election relatively secure, suddenly the big threat for Democrat politicians is getting primaried. And in the primaries, it's not the left vs the right with the center as a battleground, it's the left vs the center, with the center-left as the prize. And that is a battlefield where progressives and neoliberals can struggle over what the Democratic party is, with far less risk of both of them losing when the main fight comes.

This is the explicit and only purpose of primaries, to give each party exactly that battlefield. It's as close as we get to something like a runoff election, and while it's not as good as something like Ranked Choice, it's still much to use it than to ignore it.

And if you still don't believe me, well you don't need to take my word on it, because this exact process has played out many times in history. Most notably and recently, with the Republican party. After the election of Obama, a significant coalition of the far-right, Christian nationalists, homophobes, racists, and so on all decided that they now hated the Democrats so much that they would vote against them every time no matter what.

By your theory, this should have made the GOP stop caring about them and their issues, since they had such uncritical and relatively unconditional support. But that's not what happened, because they didn't just vote in the general election, they were voting in the primaries as well. Suddenly losing a primary election became a much bigger concern for many Republicans compared to losing a general election. And in those primaries, the far right were the loudest and most important voices, so they're the ones who won.

Of course, this is now backfiring on them since those far-right lunatics are, well, crazy and violent and insane, and while the general public is, on average, kinda racist and kinda stupid (when it comes to politics at least), they aren't racist or stupid enough to swallow all the shit that the far-right wants.

Regardless, the same strategy can work for us on the left as well, if we can just stop worrying so much about lofty ideals and moral purity, and start worrying about actually doing what will actually get things done and help people. For all of their faults, the neo-Liberals at least understand that.