Yes, Doc, it could be that heat has damaged the carbon track inside one or both of the volume pots. That can lead to some strange behaviour.
There is nothing active in this guitar's wiring, so its all electrical, not really electronics.
From the photos I'm still not 100% convinced about the solder connections to the bent pot lugs. I'd reflow them with fresh solder if it was mine. And the solder blob on the back of the volume pot in the lower of the two photos doesn't look great. Another join I'd redo. Lead-free solder needs to get hotter than leaded solder to melt and flow well, so it's not easy to solder to the back of a pot.
If you look at the second photo, the solder blob on the left has a rounded top and flows onto the rear of the pot. But the solder blob on the right with more wires is more of a round blob. This can mean a poor join, where the solder is lightly tacked on rather than making a solid connection throughout.
It could be a combination of things.
Are you familiar with Gibson-style 2 x volume 2 x tone controls?
If not, you may not be aware that with both pickups selected, if you turn one pickup off, you get no sound at all from either pickup.
Some people thing this is a problem when it's just the way the circuit was designed. The output of the volume pot is taken to ground when the volume is at 0, and as the two pickup outputs are connected together when the selector switch is in the middle position, the second pickup's output is also taken to ground, so no sound. It's one reason you have a selector switch (to isolate the output of one pickup from the other), otherwise you could just use the volume controls.