Jazz basses are great, but P basses have their uses (think Iron Maiden).
P basses tend to sit in the mix better because they have more mids, especially when strung with flats.
call me old fashioned. but find a bassman amp.. all is good
Pairing the right amp with the bass is important too. Often bass amps are intentionally coloured and it helps to get one that doesn't suck out frequencies from a place where the instrument is strong. I discovered Smith basses and SWR amps both have scooped mids, so scooped that I couldn't actually hear the pitch when playing live. I ended up keeping the bass and getting a GK RB700 instead as it was a bit more aggressive, especially in the upper mids. Both are gone now but that combo worked really well live. P basses and SVTs are a classic match, anything with a graphite neck into a valve head or SWR is great. Fretlesses need a LOT of mids to sound great (and to hear oneself). Jazz basses are pretty much the industry standard but they are particularly complimented by Mark Bass for finger style if you like that bridge burp. Or SWR if you prefer a Marcus Miller scooped snap. MM Stingrays go really well with valve heads - particularly the Mesa Bass 400+ and Bassman. But also with Warwick heads maybe because classic Warwick BASSES (like the Thumb) have a similar timbre to Stingrays.