The FAA has some really arcane rules, which is why aviation law is an actual speciality. Like, for instance, as a private pilot I can't fly a coworker with me to a conference and be reimbursed by my employer, but I can fly just myself and be reimbursed. There's a lot of subtleties to the FARs.
But if you're talking about private pilots flying people around like Uber drivers do, I can understand why they would clamp down on that.
Commercial and Air Transport pilots have much more rigorous standards than private pilots do, including stricter medical requirements and more thorough yearly medical exams. Private pilots will soon be able to get by with just an annual physical at a family doctor after their initial medical, whereas ATPs have to have a yearly EKG done, for instance. There's also more thorough training, higher hour requirements, etc.
Basically, people and equipment flying passengers are held to the highest standards (sometimes, maybe too high) which is one reason commercial aviation is so safe, and the FAA does not want private pilots who may not meet those standards running underground, fly-by-night airlines. All it takes is one crash wiping out a family in an "uber air taxi," and maybe a few people on the ground if it hits a house or something. In addition to being a tragedy the people involved and aviation in general, it would bring a ton of negative publicity to the relevant licensing agencies followed by mountains of lawsuits.
Without massive changes to the current aviation legal system, I can't see any way an "Uber air taxi" could work. Unlike a lot of cities which may not fully enforce a prohibition on services like Uber, the FAA will bring the wrath of God down on anyone they find willfully breaking the rules.
Until that comes down, jets will simply not approach uber-useful affordability for even people who buy full-fare first class seats.
(Fuel consumption is also much higher both in gal/h and in $/hr for jets compared to pistons)
And piston planes are not nearly as useful for air taxi due to them being unable to fly high enough to avoid most weather. Oh and in general turbines are an order of magnitude less likely to fail per hour.
The biggest thing that could prevent this welcomed development? The government. Specifically that the FAA is likely to clamp down on "Uber Air Taxi" market and declare them common carriers.
Got to protect those monopolies.