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What’s my control scheme here?

Replacing the capacitor with something like a .1uf will make for a more drastic shift in tone control. The "rolled back" sound will be much more muffled, like dub/motown style bass. When wide open, it's still clear.

And yeah, it's possible that the pot has kinda gone bad, so if you were going to that level of effort, I'd say replacing the whole tone pot would be a good move.

Also, with that in mind, you could get a tone pot that has a push/pull switch and do something like a "dual cap" version so you can have the traditional .047uf and the .1uf available, all you have to do is set it in the mode you like. It's not much more than a basic replacement either.
I do them in a lot of basses and no one who's tried it has disliked it.
 
So, what would your preference be for passive PJ DiMarzios and 3 control positions, and why, please? V-V-T vs V-T-3way vs V-B-T

None of the above. My fretless has a mode switch (2 positions). One is a neck only mode, and there is a tone control for that mode - a bit smaller capacitor value than most basses, but otherwise a similar circuit. That gives you a "P mode" - sounds and acts just like a P pickup by itself - there's no second tone control loading things, and the volume control (for both modes) operates like it should, not backwards like on a VVT setup (that messes to much with your tone when you're just trying to roll back the volume). The other mode uses the neck pickup as a "woofer" - it's low passed at 300 Hz. The bridge pickup runs full range, and there's a tone control for this mode as well, but it has a much smaller capacitor than you'd find in most basses - up full, the tone is what you'd expect, but "off" it essentially gives you the same sound as a hotter pickup (but without the volume change).

I don't like the blend you get from combining a P and J pickup full range - the G/D side of the P pickup is too close to the bridge pickup, so the comb filter that results from having both of them run full range is not pleasing to my ear. The standard J-j spacing (60's or 70's) works fine for me (with a slight preference for the 60's spacing)

Anyway, this scheme gives me a couple very usable sounds with tone tweakability on both, which is at least one more than any of the stock ways of doing things (to my ear). Both modes also have the same amount of low end (which the FOH person tends to appreciate). Yes, it costs a couple more dollars, has a few more parts, and takes longer to wire.

My avatar is a bass I wired for a client - it has 4 modes (It's a bit of a monster), 2 of which are similar to the ones I referenced above, as well as a couple of others. I do this kind of stuff for a few select clients in my area - I've never seen anyone else do some of the twists I do, although I would imagine someone else out there must be on the same trail.
 
Thanks @micguy - you gave what I asked for, and then some! I'm learning the science/engineering behind pickups and control circuits while equating design choices to differences in what I hear, so I can ignore all the "guitar tone voodoo" that's out there.

Do you share your designs, or are they part of your business?