The other reason [I've been pretty much avoiding Jazz Basses] is that I don't like Jazz music. I can listen to most genre's but apart from a small number of songs that may be loosely classified as "jazz" i can't get into it. I particularly dislike most live jazz I've ever listened to and find it to be too much "musical masturbation" and not enough "entertainment". (I generally think the same about 10minute blues solos as well). Hence the "Jazz Bass" name alone puts me off. Ha ha ha. Yeah..... shallow, I know.
The only reason I can think of for me to even try a Jazz is that I prefer the narrow neck to a P-bass from the quick (unplugged) play I have had of a couple. I have yet to play any through an amp yet.
The narrow fingerboard is, in and of itself, a good reason for some folks (like me) to prefer a Jazz Bass. My hands aren't all that large, so I like it for the easier reach, easier playing. Also, I've been a guitarist for many, many years, so I'm always ready for any narrower (closer to guitar) string spacing - can't
get too narrow for me! I also very much like the offset body shape of the Jazz - again, it suits me better physically than a P Bass. The Jazz "neck" pickup can give you a lot of good sounds, and is known for mixing well with acoustic guitars (listen to Mike Viscegia with Suzanne Vega).
So to me it comes down to those physical dimensions, and which suit any given person better. Some people feel more comfortable with a P Bass, and some are more comfortable with a J Bass. Both basses can be gotten with either pickup configuration (though I don't see too many P's with a Jazz neck pu), so in that sense the pickups are a less vital consideration than how the bass feels.
Thumbs down on the jazz music dis, though ... to each his own, but ... there's a
ton of great jazz out there that's
way more than just "masturbation" - you just have to have ears to hear it, same as any genre. Try some Paul Desmond for some of the most melodic riffing you'll ever hear!