Thanks for all the responses. To answer the various questions:
I don't know if it's the right foot switch for the amp - it came with the amp, but it was an estate sale so there's room for error
The problem seems limited to the foot switches, it works fine when I just use the front panel switches
None of the switches seem to be latching, they are just momentary
Our best guess is that as @Sparky Mark says, the foot switches need some switch cleaner and a bit of use
In other news, I had a friend stay over Christmas and she's an electronics engineer, so we took the amp apart to look at the front panel illumination.
Inside the amp:
Confirmation of the age:
It turns out it didn't need great electronics skill. The front panel is connected to this board, and there is a 2-pin connector on the board (yellow ellipses) which comes directly from the mains transformer (yellow arrow). So the panel AND its drive circuit have been disconnected all this time! I had worried that maybe the panel was disconnected but the driver had still been live and was possibly damaged.
I re-plugged the connector, and voila!
Mmmm!
While we had the case open, I took a photo of the power supply capacitors. There's a bit of 100Hz hum on the speakers and we're suspecting dried out capacitors. There's a bit of work to do to remove them though.