There are several facets to this issue. First, I'm a licensed private pilot. The regulations for private pilot are rather different than for ATP (Airline Transport Pilot), so the phrase "Private pilots cannot comply with the regulations of the large airlines" is somewhat misleading.
Now, the question comes to be: Is this pilot doing a "For Hire" service. This really is the crux of the issue. If the pilot *truly* is going somewhere and you want to go with them and split the cost of the trip, this is perfectly legal. However once you (the pilot) cross the line of going places because someone wants to go somewhere, that would be a For-Hire service. This gets a little gray because the pilot can't charge the cost of the trip, but must "share" expenses, legally. This can get somewhat hard to prove. However, if you are a pilot that does for-hire transport, then you must have a commercial license (not ATP), which again has some different regulatory requirements (pilot performance, medical, etc) than a private pilot.
I think I'm on board with the FAA on this one. Uber drivers are for-hire, really - I don't think anyone could rightfully say you're just hitching a ride. "Uber for planes" is the same thing.