I'm unsure if you're familiar with the experimentation that went into the development of the humble bass pickup? To me, the evolution of the bass pickup to meet such a simple design brief is really helpful background (ie. amplify the fundamental frequencies of a vibrating bass string without removing harmonics).
No doubt you're familiar with early approaches to pickup design, which were first applied to guitar, except for Paul Tutmarc's effort; he believed his attempt was just a variation on the Bell company's already patented telephone receiver.
This was when Rickenbacker/ National and the Dobro brothers were doing their whacky things in the musical kitchen with frypans and toasters ... while Gibson's homegrown electronics genius, Seth Lover, was working out how to tame Les Paul's 60Hz hum, and Leo Fender applied his single coil broadcaster/tele pickup to bass.
What was revolutionary about Fender's subsequent pickup design, which shrunk his 4-pole single coil and doubled it, was that it matched Lover's humbucker idea, wiring the two pickups together in series as a single voice. With two poles per string, this design not only removed the hum but also tamed the (then potentially speaker-destroying) single coil punch.
By producing focussed low-mids that weren't boomy, the split coil design became the golden standard for bass-string frequency response that amps and radio speakers could faithfully reproduce; coincidentally the same speakers playing the bass lines of early rock and roll.
Fender then applied this 2x single coil two-pole-per-string approach to the jazz bass, but wired them in parallel to make to two pickups work independently, allowing the bassist to dial in their sound from (effectively) humbucking to single coil.
(Years later Fender Corp. reissued the single coil "Tele bass" - amps now able to keep up - then poached Seth Lover from Gibson to design their Wide Ranging Humbucker which replaced the single coils in Tele basses from 1972.)
Meanwhile Grateful Dead's Alembic crew had been working on how to actively shape tone by making their pickup circuits electrically active. Leo Fender applied this approach when designing the pickup for Musicman's new Stingray bass. He developed this pickup with two separate coils - each with one large pole-per-string - that can be wired (switched) into various configurations (humbucking/ single coil; serial/ parallel; active/ passive etc) for a range of sounds.
If we include the experimentation of Spector / Steinberger in using EMG's developing active underwound pickup in this list, as well as finding piezo pickup applications for bass; its easy to appreciate the range of experimentation that has resulted in the fairly stable tribes of pickups available today.
Add to this several generations of musicians each trying to replicate their preferred bass tones which, on balance of probabilities, were played on a P or a J, and you end up in the place you first described.
At the end of the day, while the ears might hear, it's the mind that determines how desirable a sound is. This is also the likely reason snake oil works better on guitars than basses.