It's kind of an accepted convention that the sitting President gets a clear run at a second term. So once Biden indicated that he didn't intend to step down, that was pretty much the end of it. The one thing the Democrats have over the GOP right now is that they're essentially still functional as a party, so the last thing they want is a potentially nasty and divisive leadership campaign to expose the cracks.
Also they don't have a lot of obvious candidates for succession. Kamala doesn't perform terribly well on the public stage, Pete Buttigieg polls at virtually 0% among black voters and Gavin Newsom is seen as a bit of a west coast liberal elite, which wouldn't play well in the swing states.
As for Trump, as far as the MAGA loons are concerned he IS the Republican party now, much as the establishment probably wish they could get rid of him. There's no closing that Pandora's box until he drops dead.