@RobbieK's video demonstrates that the capacitor plays no roll at all until the tone pot is dialed almost all the way back. He replaces the cap with a piece of wire, and the tone control works fine until close to the end of its rotation, when it suddenly turns into a crude volume control. The only role the cap plays is as a sort of dam that only allows the highs to go to ground. (A secondary effect of the cap is the resonant peak it introduces when it comes into play.) So for most of the tone pot's rotation, it's just the pot's parallel resistance dumping highs to ground.
It's the removal of the pots that makes straight-to-jack wiring brighter, because even at 10, a pot is allowing a bit of the signal (treble) to bleed to ground. (In other words, the presence of the cap is irrelevant to the circuit--and thus the sound--when the tone control isn't turned way down.) When the pickup is wired straight to the jack, hot and ground are completely isolated from each other, so the full signal makes it to the amp and the pickup's resonant peak isn't damped at all, which is why it's generally perceived as rather harsh and brittle sounding. This is assuming a fully passive system, of course.
There's more discussion on the subject in this thread: Capacitors and Tone
And to this thread's larger point, although I do use the volume and tone knobs on my (single pickup) bass, I'm drawn to the idea of a bass with no knobs, partly for aesthetics and partly for an acoustic instrument-like simplicity. Rob Allen has an interesting approach on some of his models: a volume knob accessible to the player and a trim pot for the tone that's accessible from the back that's meant to be more set-and-forget.
It's the removal of the pots that makes straight-to-jack wiring brighter, because even at 10, a pot is allowing a bit of the signal (treble) to bleed to ground. (In other words, the presence of the cap is irrelevant to the circuit--and thus the sound--when the tone control isn't turned way down.) When the pickup is wired straight to the jack, hot and ground are completely isolated from each other, so the full signal makes it to the amp and the pickup's resonant peak isn't damped at all, which is why it's generally perceived as rather harsh and brittle sounding. This is assuming a fully passive system, of course.
There's more discussion on the subject in this thread: Capacitors and Tone
And to this thread's larger point, although I do use the volume and tone knobs on my (single pickup) bass, I'm drawn to the idea of a bass with no knobs, partly for aesthetics and partly for an acoustic instrument-like simplicity. Rob Allen has an interesting approach on some of his models: a volume knob accessible to the player and a trim pot for the tone that's accessible from the back that's meant to be more set-and-forget.