Well, I'm narrowing the problem down, at least.
I tried wiping my personal.ini completely, no effect. There's no username and password in my settings.
But then I finally had the idea to actually open one of the epubs I'd downloaded in the course of all this testing, and there it was - the cover was downloaded and stored inside the epub just fine. I'd be more embarrassed about that oversight if I'd had time to do any real reading in the past month.
Anyway, going "edit metadata in bulk -> change cover -> set from e-book files" causes Calibre to get the cover from the book as you'd expect. So whatever's going wrong for me is just in the step where Calibre's displayed cover is updated from the downloaded epub. That's good, it means there's no data loss and I won't need to update the whole stash of books I've been accumulating for my next reading spree. I've tried various permutations of "Update Calibre Cover (from EPUB)" and "Generate Calibre Cover" FFF settings without getting any covers showing up in Calibre, even with both set to "always". Very puzzling, I'll poke at it some more and see if I can figure anything else out.
I was at version 3.30.0 of Calibre and tried updating to the latest 3.32.0. Maybe I could try downgrading to an earlier version from over a month ago to see if that's part of the problem.