When the Play/Pause button is held down, a "good" unit misbehaves in exactly
the same way as the "bad" unit, whether or not the remote control is
installed.

The Play/Pause button isn't jammed, so it would seem the fault is somewhere in
the wiring.

I have the service manual. When I find it, I'll tear into the lid (which holds
the key pad). It couldn't hurt.