The actual computer part of the SAT-NAV is as you suspect in the glove box.
The problem usually is that the person disconnecting the battery does not allow for enough time after turning off the ignition for the computer to correctly power down.
I have never heard that turning the lights on and off can help . (navi computer - cd reader has even no lights signal connected :D).
But you are right with your "not enough time for power down diagnosis. Is is very common failure.
Memory content is damaged after this and navi reader will not start anymore. But this is not the situation described by author.
He's navi reader is still working because shows logo or time when pressing eject button.
So the problem is with control module (this with buttons, mounted over radio). It is not sending proper ON signal (+12V) when card is inserted into card reader.
There can be some reasons for it:
1. control module damaged after battery disconnection and connection (radio control satellite could also not work then).
2. fuse under ashtray burned
3. cables problem between reader and control module (multiplex network or turn on/off signal).