Just as a follow-up, I ordered a replacement drive chassis (a plate with the optical block and spin motor on it) off ebay the evening I posted that message. It arrived yesterday, I installed it, and it worked.
I'll give points to Sony for making such a modular design (the original "fat" model was rather more difficult to mess with), but I would have preferred to get a bit more life out of the thing than I did. I'd bet if I plugged in my first CD player (made around 1985, though I haven't used it in years), it would still work just fine.