I wonder if water has penetrated the switches?
I would expect everything else on the board to be impervious to water
Yes it will have.
Immersing the circuit board in pure (99% or better) IPA should drive out the water. It also evaporates quickly and leaves no residue.
The switches can probably be de-soldered and replaced fairly easily. If it can be established that the switches are the problem.
The gentle scrub first with a toothbrush and IPA should clean off any oxide that's shorting out anything else.
The 3V battery cell will be discharged and needs to be replaced.
Bombay Sapphire or Gordon's won't work as it's 60% water.
The solution that Bigiron used to improve the sound of his Bay City Rollers records was likely diluted with 50% distilled water.
Proper methylated spirit is worth a shot if no pure IPA is to be had. But might contain impurities and leave a residue.
However nobody should invest too much cash into this. It might not work if one of the components and/or the board is corroded or damaged.