Back when I owned a 4003 bass, I reckoned it sounded like a classic Rickenbacker bass even unplugged. It had that mid-scooped sound with a rock solid fundamental and treble sizzle. The Rickenbacker pickup placement and design, perhaps mostly by accident, enhance the best elements of the tone that all that maple, plus the through-neck, impart.
I think that particularly in basses, the neck rigidity accounts for a lot of character in the tone. I currently own two Fenders with fairly rigid necks and they sound good. I've played a 1977 Stingray that felt completely rock solid. There was no flex or give in the neck at all. I've made a few parts basses with MIM Fender necks and the worst could easily be flexed by hand, dropping an open E down to an Eb by pushing the neck forwards.
I don't think that pickups can compensate for a body wood and construction type that either promote or rob certain frequencies in areas of the sonic spectrum. Putting Fender-like mids back into my 4003 would have required careful EQ, multi-band compression etc, rather than simply chopping a Dimarzio Model P into the "sweet spot".
