TL/DR: Both are great, but PJ is more useful on the bandstand.
I have both, and love 'em both. But I'm more pragmatist than purist, and it's a pain and a delay to be changing basses while dancers are standing waiting for the next song. With the PJ you can go straight P all night if you like, but you can also make easy, quick adjustments on the bandstand and even mid-song when needed. In some venues, it's the bridge J pickup that rescues my tone from the mud.
With the PJ you can quickly and easily roll in some bridge to cut through and lend pitch definition in a muddy/boomy room, or tighten up/define the individual notes of staccato funk or 16th-note boogie patterns that turn to mud in some rooms (Just ask Rocco--he uses the J bridge for what he calls "seasoning") Go all the way to the bridge for a Jaco-like tone for soloing if you like, though I don't solo.
When you want a P, turn off the bridge pup. There's no reason you should get added hum if the bridge PU is off, and no way for the J to "taint" or "spoil" the purity of the P when it's off (I know, electronically it's not quite exactly the same, but my PJs sound every bit as P-like as my straight P, and I doubt anyone could pick out the PJ in a blind test.)
I haven't had any issues with fiddly-ness, but straight P is always my starting point. I dial the J in when needed and back out when not.
Don't know why you wouldn't be able to get a "good sound" out of a PJ, (assuming you consider a straight P a good sound) or be stuck with added noise, as you can aways turn off the bridge pup and make it sound like a straight P, and the majority of PJs have a noiseless J in the bridge anyway.
If you want an excuse to bring an extra bass to a gig, then by all means get one of each, but switching basses mid-set is the kind of fiddling I'd rather avoid if possible, and that's why I've always liked a pj (or, these days, a 3X pickup setup with ser/par switching which gives you PJ+ JJ + MM).