Had Leo had invented the Jazz bass first, with Jamerson and others having played one of those instead, we probably wouldn't be having this discussion..
Jamerson and other have had the Jazz available, because the first split coil P-bass came out in 1957, the first Jazz Bass in 1960. Jamerson switched to the Precision Bass in the early 60's with a '58 or '57 that was stolen, and then bought a '62.
if he ever wanted, he could've been playing the Jazz Bass, as many did.
both the "split coil Precision" and the Jazz Bass were new at the time.
I thing I've learned in 20 years of playing, is that when you go into a studio for recording, 99% of the time they ask you for a Precision Bass... I went in studio with Jazz, Stingray, different Yamaha (mostly with a split coil, however), Gibson, Aria SB1000 and others, but they always preferred the Precision, of which I always have one, even in those periods when I don't "like" the Precision.
last experience was a few months ago: I was recording for a punk band, for the new album, and live I was using my Yamaha BB3000 or a Manson JPJ Signature (EMG 35P and EMG 35TW).
I was using these two basses because they gave me a lot of "growl" with a pick, sparkle in the hi-mids and highs, a metallic tone that I was liking a lot.
it was a period in which I didn't have a Precision, because I was selling-buying instruments and my main basses were a Yamaha BB3000 and a Manson JPJ Signature.
the Manson, while live sounded good, recorded was lacking a lot of "punch" and frequencies, and didn't work with the guitar.
the BB3000 was something nearer with the split soloed, but we had to work on the starting EQ to add and remove frequencies to make it work better.
the producer repeated me a lot of times: "if you've had a Precision, it would've been easier".
the mixing took a lot, it's still not finished, and we had to work a lot on the final bass sound, because it just didn't fit.
PS: the Yamaha BB3000 doesn't sound like a Precision even with the split soloed, it lacks the low-mids of the Fenders, and has far more highs and basses. the overall sound of that bass is nearer to a Spector than a Fender...
the Manson has an EMG 35P, which is a reverse-P in the soapbar cover, and the sound (also with mahogany body, maple neck, ebony fretboard and maple top) is nothing even near a Precision
those basses work wonderfully in live situations, they give me plenty of options, but in studio not that much.
a Precision in a live setting is easier to be heard.
OBVIOUSLY a lot of players prefer modern sounds, sparking bright instruments, metallic tone, scooped mids, chainsaw sounds, piano-like tone, sequencer note quality.
but then, in a studio when you have to record for others than your specific band, probably the P bass is what they want.