The numbers I gave were national averages for the US. Of course those numbers will vary by location.

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.

Rural areas are more likely to be recreational plus agricultural, due to proximity of sophisticated clients in big cities.

If someone's traveling to/from a work purpose, why wouldn't a corporate jet flight be categorized as a business service?

Taking that same jet to ski in Tahoe? Personal/recreational. Taking it to a business meeting with a customer/prospect? Business.

A lot of what people think as "Silicon Valley" is actually centered around general aviation industry - for example San Mateo County at least in 2015 had majority share if its economy based on aviation, with majority of it being GA

Going to need a source for that claim - generally the largest industry in a given area is government, then hospitals or schools, then some specific niche. I don't think there's any airplane manufacture in NorCal anyway - this would be a bit more believable about Wichita.

Because you're looking for "airplane manufacturer" and possibly large one. Though for example just a stolen throw from Google Mountain View campus you have Northrop and Lockheed-Martin.

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)