Starmer was deposed by his own party, who saw him as a liability. That's rather different to the Democrats trying to compromise with the Trump "administration" [1]. They'll probably try, and actually I'm not sure whether or not that's a good thing for them. But I doubt it will make any difference whether they do or be as obstructive as possible. Even if they do manage to strike some deal (Trump does seem to be obsessed with "deals", doesn't he?) he'll renege on it. I doubt the implications of Trump/Republicans thwarting every attempt at proper governance will be lost on those who just want some sort of functional government, and there is a significant fraction of people who want just that. Plus of course it gives them a convenient scapegoat, and doesn't everyone like a scapegoat?
So we'll see. I think the democrats will have to try hard to snatch defeat from the jaws of victory, especially with the prospect of Trump becoming ever more unhinged. Narcissist that he is, he's already lashed out at the Republicans who he perceives as failing him, as witnessed by his blaming Greg Abbott for losing Texas. In other words, Trump is very much a Republican problem.
[1] Maybe we should be saying "AINO": Administration In Name Only?