I think times are a changing - it is true that many television appearances in the UK (though by no means all) have either Jazzes or Precisions - that they are often fairly inaudible in modern times is also a point noted by many in Basschat in respect of a regular UK live music television show.
However the basses I see in the hands of local pro and semi pro bass players is completely different - Musicman or SBMM are popular, Rickenbackers are far more popular (and sound far better and varied) than you would ever expect based on posts on Bass forums so it is a much broader spread.
I'm afraid that I have been completely sold on a fat, focussed bass sound a la my heroes like Bernard Edwards, Pino Palladino and the like.
I play in blues bands with a hint of R and B, funk, jazz and rock and am lucky enough to be able to choose which bass I take to each gig (mine are all Musicman of one sort or another) - there's no 'I don't like red basses because they remind me of communists' or other idiosyncratic control freakery bordering on personality disorder issues

drool
As its that time of year with sunny afternoons and evenings, outdoor gigs playing through large PAs to larger audiences I'm starting to make different choices including a Bongo a couple of weeks ago (sound guy came and asked me what bass and model it was it sounded that good!!). Anyway last weekend I made a very bold choice (I thought) - Stingray Fretless - I apologised in advance to the sax player stood next to me on the outdoor stage for the possibility of nearly (or barely) in tune playing - and let rip through my LM3 DId into a very large PA - the DI button carefully set to 'post EQ' - there's no way I'm letting the sound guy remove my mids whilst I'm playing Fretless - well at least not before he's received them!!
Anyway the gig went very well - I had forgotten there are a coupe of songs with slapping and the Fretless Ray sounded very similar to a fretted one when slapped - so no problem there. After we'd finished the sound guy (actually also a good bass player) said the bass sounded absolute quality - all Paul Young at times (he was referring to Pino's sound of course).
So there you go - the Stingray remains a great and versatile bass capable of a 'quality bass sound' to quote the independent (of our group) sound man - I've known that since the late 70s when I first had one!!
I don't think they've particularly lost popularity - if anything the new Specials sound like they've stoked a lot of interest - I'm looking forward to playing one of the new Specials when they arrive here - I'm sure they will be equally good if not better - that Fretless sound was so good last weekend though!!