The estimate is that about 65% of general aviation traffic is related to "business and public services". [1][2] The other 35% is personal in nature. From that, about 22% is flight instruction for new pilots - most of whom are trying to get into the airlines. [3]
So that would mean about 13% of general aviation is recreational.
(Also, for what it's worth, many general aviation pilots I know aren't rich by any stretch.)
[1]: https://download.aopa.org/hr/Report_on_General_Aviation_Tren...
[2]: https://www.govinfo.gov/content/pkg/GAOREPORTS-GAO-01-916/ht...
That said... you'd be surprised how many flights out of the small GA airports in big cities aren't personal flights. Air ambulance, news crews, law enforcement, firefighting (usually more on the periphery of the city), utility companies doing inspections, and aerial photography all tend to operate from these locations.
On the other hand, I'd certainly expect the number of flight school operations to be higher in areas with a large population.
Taking that same jet to ski in Tahoe? Personal/recreational. Taking it to a business meeting with a customer/prospect? Business.
I can't find the original webpage after 7 years, but the source was San Mateo County page related to county-owned airports, and it included all sorts of aviation industry (both flying and ground side)
Not to mention medivac flights, overnight delivery of many laboratory medical specimens from doctors' offices, law enforcement, high-quality aerial imagery in maps apps, farming, access to rural communities in areas like Alaska...
I believe the typical American who sees the upper middle class, the same way the rest of the world sees america as.
For reference - considerable portion (possibly even majority, especially if you don't count USA due to different licensing) of airline pilots will have their first jet/turboprop experience when they will be training on airliner after getting their "frozen ATPL". In one airline it was common for chief of training to give every new trainee a copy of MSFS 2004 with professional addon for the first plane you'd graduate for, and the better trainees used that to become familiar with plane - cause usually the largest previous plane they flew would be a small twin piston plane like Piper Seneca.
For non-military pilots, most of those hours come from general aviation aircraft.