New intel: I popped the cover off the fan and plugged it into an extension cord, and the fan doesn't run. The extension cord, to be extra-sure, does power, e.g. the vacuum cleaner I used to suck some of the accumulated dust and gunk off the fan.
There's some blackening on the housing above the motor, too, although to be fair that could have been there for 20 years, since I haven't been in there since we bought the house a couple of years ago. But that's some pretty strong circumstantial evidence that the fan motor is the problem, not the new switch.
So I'm off to the hardware store to invest in a new fan motor. Will advise.
But in the meantime: Any idea how putting in an inadequate switch could have blown the fan motor?