Doesn't a post like this pop up every few months?
Considering the depth and breadth of recordings (live and studio) of Jazz Basses in a band setting, the Jazz Bass itself can cut through the mix quite capably, thank you very much.
Even considering that, we must accept that the Jazz Bass is a different animal. The 2 pickups and their locations promote the familiar 'scooped' sound, which is good for some things, not so with other. Lowering the volume on ONE of the pickups will change the "character" of the scoop. I hope someone better versed will chime in with the "phasing, filtering, combing, rinsing" of what I just referred to as "character". Personally, I find the Jazz more nuanced, whereas the P or PJ is a blunter, more "here I am and this is my sound" kind of bass.
Bottom line: If the sound in your head is a P, do not expect that from a Jazz, or vise versa. The other day, a friend of mine was recording at his house and I noticed his EQ had a very sharp scoop. Since I saw he was using his favorite P bass, I told him "Why don't you try the Jazz and set the EQ flat?". Indeed, that was the sound he was looking for, on that particular track.
YMMV
IMHO
Diego