Something sounds weird here.

Since when does removing the tone cap cause no sound?
You should be able to remove all the tone circuit (i.e. pot and cap) and still get sound, more sound in some cases.

Secondly, I wonder why it wasn't soldered to the pot like every single other guitar I've ever seen, which is not many I'll grant you, but it seems like it's the norm.

If it wasn't a tone cap, then what was it? (.01uf does sound like a likely tone cap for humbuckers)

"The other wire on it just rests between the metal of the lower string bridge and the body of the guitar."

It sounds strangely like the grounding wire that typically runs from the scratch plate to the trem claw (in a strat style guitar) had a cap in series, which also makes no sense to me. Even so, removing that connection should get you hum + guitar sound, not hum only.

Bizarre I tell yah, just bizarre.