Well I have to admit, when I'm playing funk that sounds like it's from a cheezy low budget 1970's porn movie, the first bass I reach for is my Jazz. Not only can I get all the nasal tones I want, but as you note it's capable of so much more - like bark and honk.

Seriously, I actually cut my teeth on a Jazz bass. This was my go to bass for well over a decade...
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Even when I wandered from Fender Jazz basses I still made sure to get something that had at least one Jazz pup...
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Then this happened...
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And that led to this and this...
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Which led to a bunch more of the same.
The journey isn't over yet, but I can honestly say for every single hour my "has been go to Jazz" is plugged in my Gibby's (birds and SGs) get hundreds of hours of amp time.
It took me a long time but I finally figured out I'm just a Humbucker kind of guy, single coils just don't excite me much these days, and generally speaking that to me is the significant difference between Gibson and Fender basses. Two very different approaches to making great bass guitar sounds. In the right hands either one is capable of creating bliss, in my hands it's a Gibby or similar Humbucker equipped bass. YMMV