Thanks Diana.McCall. I'm still trying to find good ways to check for interference; last time I looked into this, I only found programs that could work on my computer, but since that's a desktop, I can't move it around to check how network varies. (I've got several iOS devices as well, but no Android.)
Eventually it started working. After turning everything I could think of off and on repeatedly over 20 minutes, and failing to improve the situation (actually it made things worse - it all started from Plex not connecting, and went to no wifi at all), I went online for a bit to look for solutions - which was when I posted above. Thirty minutes later I tried again, having changed nothing, and it was working again. I have no idea why.
I do suspect interference is the problem, but if that's the cause, wow, is it ever a problem. My desktop and iPad are both wifi-only and I've had vanishingly few issues over the years, but I seem to have problems with the Apple TV every other time I turn it on. I'm in a high-rise apartment building and the Apple TV sees five or six wifi networks. There are apartments directly above and below (with microwaves?), and I'm starting to suspect the little beeper we use to activate the elevator also breaks the wifi connection (the elevator shaft is directly behind the TV).
All that being said ... I'm confused about what's going on when the ATV sees the network, and I enter the password, and then ... nothing. Sometimes it just returns to the password entry screen. No error message or anything. It still sees the network. Would that be interference or some kind of bug? I at least expect an "Incorrect password - please try again" or "Connection cannot be made" (which I do sometimes get, but not always).