Very interesting. I never thought about what would happen if I didn't solder the cap on my tone pot. I read that it makes your guitar a little brighter, but you cannot control the tone knob.
I have an old Fernandes that sounds nice and bright (like a metal guitar) and I don't know if it's because of the pots and parts or if it's because of the wood and neck, etc. My Strat is way more warm and less bright, and can't really dial that out by swapping pickups as I've learned. I'm a student to all this stuff.
You find all sorts of weird things. Unexpected treble bleeds, etc.
You can take and just go through different cap values of the same cap type and there is a tonal difference. Like .015, .022, .033, .047.
Higher values make the sound darker, warmer, rounder.
Habe a strat I hate with the setup with a standard .047 due tonhabing A3s which can be dark on top. Made it sound like a bass guitar.
Went back to my lower value cap, and there the sound was!
Throw in that each cap type has its own sound, and there you go.
Yes, just as mentioned, you can also remove the circuit entirely for a different sound as well.
Even when the tone knob isn't engaged, the cap is affecting the high end frequencies and certain resonant frequencies of the instrument.