Russel,
You are completey correct except it is electrolitic capacitors that fail, and they are in the tuner also, it is rare that the entire amp fails in one whack, the fuse in the tuner could blow, and yes if it had a CD the cables could be loose. The control head usually indicates faults ( but not bad "bose amps")
The pse also controls the lumbars, trunk lid handle, vacuum at rest for the HVAC and the back up rods, if those dont work it could help trouble shooting.
If the system senses a leak it also shuts down, but after resetting you should hear ir run. There is a 10 second shut down in locking and a 60 sec shut down in consumers.
I would try the keys since they control the pump directly, whereas the remotes interface to the pump also