I'm not sure I see much of a difference. Most guys seem to get by in both playing on a standard 4 or 5-piece kit. Both Buddy and Bonham played essentially the same drums their whole career, and Weckl, Cobham, and Chambers have all gone through their huge kit phases.
I don't think the popularity of the genre is what is driving drumkit sizes, as much as what the drummer wants to play.
I think there would be more studied players out there. But if you had that, I think the musical gear industry wouldn't be as big as it is now (a large percentage of sales is generated by people who would like to get their feet wet, and will just go out and buy something). If jazz was the predominant form with more learned players in it, you would lack the "Joe Lunchbox" population that would make up a majority of sales and prices wouldn't be as cheap as they are.
But then again, there'd be a lot more pretense out there too. So there would be that to contend with.
The question is too general. First of all, what kind of jazz are we talking about? In the pre-war swing era, jazz was the popular music of the time, staging could get elaborate, sound reinforcement was embryonic, and drums needed enough loudness to keep up with large brass sections. So it was not unusual to see kits with gimmicky components, and stuff like tympani's and gongs. But other popular bands had rather modest kits, specially if they were on tour. So IMHO, popularity is not a determining variable.
Almost all of the drummers in the indie-rock scene are on 4-piece kits, or at least that's how it seems to me. My son is in a signed/touring indie band as well as a couple of other band projects, and all of those guys are using 4-piece kits, and so are most of the drummers from the other bands he tours with.
I don't know if this is due to convenience (there's a limited amount of room in the band trailers and vans) or if it's from the standpoint that they think 4-piece kits are hip and cool. Possibly a bit of both.
I know that in my own efforts as a drummer, I started off as an avowed 5-piece kit player, and wound up carving it down to 4-pieces for a few reasons:
1.) fewer pieces to cart around/easier to pack and load. (Yes, I realize the difference was only 1 drum)
2.) I liked the setup a bit better from an ergonomics point of view - I liked my ride placement better.
3.) Most of drumming I was doing was for churches, and most of the house kits were 4 piece, so I practiced as I performed.
I know that's a bit off topic, but I'm not sure that genre really defines how big or small a drummer's kit is, or at least not that much. There are, of course, exceptions - it's pretty tough to cover Rush on a 4-piece with a single kick pedal.