I played a Jazz Bass for about 10-years before I ever picked up a P-bass. I was immediately put off by the wider, fatter neck, to the point that if I ever bought one, I'd get a Jazz neck to put on it.
However a few years ago my grandson down it Australia got a Squier Affinity Precision Bass Kit for Christmas and we were there for it. I had to try it out of course, but was surprised to find that it was fairly comfortable for me to play. Come to find out it was still a wide neck, but it was a slim, wide neck. I would be OK gigging a P-bass with that neck on it, but I still prefer the sound of my Jazz.
I have a passive Jazz with 70's era pickup spacing, which puts the bridge pickup almost 1/2" closer to the bridge - 0.4" closer to be specific. That makes my '82 Jazz that I bought in 1985 sound a little brighter than my '65 Jazz that I bought in 1966. It also has better sustain than my '65 did. I run everything all the way up (all controls dimed) and with my Eden amps just use a little boost up to about 1:30 at either 250Hz or 500Hz depending on the amp. All other EQ's are set flat. With my DNA-800amp, I also boost the bass up to about 1-o'clock because that bass EQ is shelved at 35Hz compared to the Eden amps where they are shelved at 30Hz. That little boost in the bass makes my DNA gets the same sound like I get from my Eden amp with the bass set flat.
***NOTE*** If you boost your low mids too high, your sound will honk at you. If your sound honks at you, just slowly reduce that boost until the honk goes away - you'll hear it. Then note where that boost is and use that same setting next time. It doesn't take much boost, but it does require a little bit in the low mids (anywhere from 250Hz to 550Hz) to get the presence and definition you want when playing a Jazz Bass in a group.
After using round wounds (Dean Markley-Blue Steel and then DR-Black Beauties) for 35-years on it, I switched to LaBella White Tape-wounds about 4 years ago and prefer them. Unlike most of the flats out there, these tape wounds still have plenty of high harmonics. They are not quite as bright as either of those round-wounds, but are much better balanced from string to string. I really like them and the sound I get from them.