Thought of another thing to try.
If you think the switch might be the cause, you could disassemble it partway and clean the contacts, and touch up the solder joints on the two connectors. Sorry these pics aren’t the clearest.
Here’s the switch with the cover removed:
Pull off the clear light diffuser straight up and set it aside:
Pull (peel) off the rubber switch contact piece, and clean both those contacts and the corresponding ones on the circuit board with electronic contact cleaner or alcohol. I like to use something that will not only clean the contacts but leave a little film of protection to keep them from oxidizing:
When you reinstall the rubber piece, make sure the little rubber nibs* seat all the way in the two holes on either side of each contact. The rubber edge should be flat to the circuit board and not lumpy anywhere.
It would be a good time to re-flow the solder on the contact pins. (If you were to be replacing the backlighting, the correct way to do that is to desolder all those pins, since they’re what hold the PC board to the switch body and contact pins.)
* I wanted to say “teats” instead of “nibs” but I know how the prudes on this forum would react to that. LOL