sorry to bust yon bubbles but pickups do not produce sound; they process it.
It's far easier to find tone in strings than copper windings.
No, they re
produce the tone of the bass. Pickups can be a faithful recreation of the acoustic tone of the bass, or they can be a Gibson EB-0 pickup that gives you nothing but mud. Why does a P bass sound different from a J? The pickups.
If pickups had no tone of their own, no one would buy replacement pickups, since they all sound the same. Right?
Without pickups your solid body bass wouldn't make too much sound. So the pickups are producing the tone you hear through the amp. The bass, strings and amp also play a roll.
But P basses are not all that dark sounding. They sound dark on old recordings because that was the tone people wanted, and the amps back then were very limited in frequency response.
Here's the real sound of some famous Motown tracks isolated.
Link Removed
Pretty clangy, and with flats no less. Of course some have foam mutes too, and dead strings. Once they are in the mix, you don't hear the treble so much, but it makes the tone more percussive and it stands out in the mix better.
If you want a fat tone, don't use a P, use some kind of soapbar. Humbuckers (not the MM type) are fatter.