Three weeks later, I bring it into an Apple store just to see what they have to say, and of course it powered on once the tech plugged it into a power source. And then of course he says "oh, it looks like it's working fine."
So the fix was to let the iPod sit for however long it takes for the clock to drain a fully charged battery (about a week?), then it will power itself completely off. Better than trying to take it apart and unplug the battery -- whoever in their crazy mind would do that?
[robg adds: Leaving a stuck iPod unplugged for a while may or may not help. As a method of last resort, it's certainly worth a shot, though.]

