Perished rubber belt somewhere, most likely on mode-
switch motor.

If I remember this model correcly, the mode switch is
activated by a tab that travels along the underside of one
of the cam gears. There is no "mode sw motor" involved here.
Be careful as it is easy to miss the alignment of this tab.

"Since we last spoke", I found that the capstan isn't turning. (The pinch
roller is tight against it, but the tape isn't moving.)

This is an almost-beltless transport (at least, as seen from the top). The
belt on the loading motor (?) seems to slip when it's time for the capstan
to start moving, but moves at other times.

Further thoughts?