I've never had anyone question me playing a Jazz bass. It'd be tough if they did because I'll play what I feel comfortable playing. I can get a tone that works but also, as I get older my hands don't stretch as much as they used to. One finger per fret is very difficult and I find the extra nut width on a Precision harder work.
As others have said, you have to tweak a Jazz more, but that also gives more options, especially with the bridge pick up.
As for live sound, that's a real bug bear. Not so much for me, but as an audience member. Sound engineers seem to think the optimal mix is to sound like a car fitted with a huge sub in the boot, with windows wound up, parked 100m away.