I won't re-read all 7 pages of this very interesting thread, but I'm not sure if the main discussion was concerning FENDER Jazz basses, or the J-Bass configuration in general.
I've never owned a Fender Jazz; but have owned 2 very high-end J-Basses, one active (Sadowsky NYC); one passive (an Alleva-Coppola). Needless to say, both of them were outstanding instruments; top-rate, but I never felt that I was getting what I wanted from them.
However- I now own (3) J-Bass style instruments, 2 active- A Sago 2-band, and a Dragonfly 3-band; as well as a Sugi Nightbreeze passive.
I would NEVER let go of these instruments; they give me exactly what I want and more. In addition, I also have an exceptional Stevens Slant 4, which is essentially a J-Bass, but the pups and switching are so advanced and unique, its really a "step above" the Jazz bass build.
I'm not sure I'd bond with a Fender Jazz, even a higher-end or Custom shop model, I'd probably rather go for a G&L, Lakland, or Lull, and a Suhr is also on my"gotta try" list of Jazz basses.
There are many builders who make an exceptional J-style bass, I've been looking at Maruszczyk for example, Atelier-Z makes awesome instruments, Mayones comes to mind as well as... the list is long and tempting.
The fact that I'm loving my Japanese J-bass builds, and that the Stevens is amazing make me think that its the electronics and build elements that either make or break a Jazz-style bass.
Maybe Fender falls short in that for the OP and others, then again many players KILL with a stock Fender Jazz- I can recall a clip of Anthony Jackson playing a J-Bass (4-string; stock I believe) and its absolutely phenomenal. I've seen Christian McBride live using a Suhr I think, and needless to say- he was incredible. And Marcus Miller- 'nuff said.