Every time we see a recital given aboard any ship or star base, it's almost always a 'classical music' recital or performance, never a rock concert. The only exceptions to this musical bias are Riker's penchant for Jazz and Zefram Cochrane playing 'Magic Carpet Ride' during the 'First Contact' launch.
Roddenberry had a vision of an 'enlightened' humanity, and as a strict matter of keeping up appearances, it's a lot easier to sell people on a post-scarcity technologically affluent culture that listens to Chopin over Chuck Berry. It's a bit of an oversight in the long term, because a person's musical tastes aren't necessarily a pointer to their intelligence, but it's such a TV trope to assume intelligence begets classical taste in music that I suspect that has more to do with it than anything. It's a lot easier to sell the idea that music which has already withstood the test of time (Bizet and Berlioz were 19th century composers, both of which are name-dropped during First Contact) has "made it into space" with the rest of us than, like, Dion DiMucci. In fact, there's a brief quote attributed to the creators of Star Trek who state that it is "more believeable" that classical music continues forward whereas contemporary or popular music ebbs, flows, and eventually fades away.
More of an observation on my part -- I'm going out on a limb here, but I would hazard a guess that a utopian society is a lot less likely to identify with some of the darker tones of popular music from the last 70 years.
It's not as if people stop breaking up with each other or anything. But there are also certain socio-economic circumstances which are more easily expressed in the context of blues-inspired rock music than in a society where no one gets sick or loses their home because they got fired.
"Woke up this morning feeling exemplary --dun daaaa dun da dunt--
My car started right away because everything works perfectly --dun daaaa dun da dunt--
Still happily married because the hardship of financial difficulty never drove me or my spouse to seek the comfort of another person's embrace --dun daaa dun da dunt--...."
There are a handful of indicators towards such an attitude. The small talk between Jellico and Riker concerning the presence of a trombone in his quarters is a little telling of the remnants of 'snooty' early TNG's vision of humans -- we're all high-concept highfalutin brainiacs that immerse ourselves in the high points of art, philosophy, and use phrases like "Is it not the height of hubris..." (Really though, the reaction and body language Jellico employs when Riker says he's a jazz musician -- 'ohhhh ....jazz..... okaaay.')
You're correct that classical music does seem to be over represented, especially on the Next Generation. However, there are other times when characters do show a taste in alternative genres.
As you already pointed out Riker likes jazz, and is seen on more than one occasion to be performing with others on the Enterprise.
Worf is a fan of Klingon opera and the Doctor on Voyager is also an opera aficionado, albeit that of a more Human origin.
Multiple members of the command staff on Deep Space Nine are seen to enjoy Vic Fontaine's singing, which Memory-Alpha describes as "vocal-jazz and cabaret".
However, there are only two examples of any sort of rock and roll that I am aware of in the 24th century, both in Voyager. The first is in the episode Vis à Vis during the opening, when Tom is working on a car in the holodeck there is "surf rock" music playing on a radio. The second is in the episode Homestead, again during the opening there is rock music playing on a jukebox in the mess hall during the First Contact Day celebration.
No one likes to remember it, but it happened.
The third season had some issues.
There is one other out-of-universe reason for this that's worth noting:
The entire recorded history of Rock and Roll is under copyright. Any use of a song which the audience would recognize could cost a potentially significant amount of money. Better to just restrict any 'historic' music that characters are fond of to works in the public domain that can be used cheaply and without the headache and hassle of all those licenses and publishing rights. If you need evidence of just how much of a headache all of that can be... look no further than what happened to WKRP in Cincinnati.
C'mon, be reasonable - it's pop music from 300-400 years in the past! Diegetically speaking production issues aside), how many people do you know who listen to 16th century popular music? Anyone? Renaissance faires excluded.