This is not correct: William Gibson has warned about that as early as 1993: "The future is already here — it's just not very evenly distributed." [0]
If personal flight starts as a luxury good to pay for research and development, as those innovations become widely known the price should come down to make it accessible to everyone.
Rich people will also have professional pilots anyway.
The original flying car was the light helicopter, particularly the designs by Robinson Helicopter in the 1970s. The idea was that upper middle class professionals (doctors, lawyers, maybe computer programmers, etc) would prefer to learn the helicopter and commute into the city from the countryside than drive to work. This idea didn't pan out, because many/most professionals prefer city life (once the race riots stopped in the 70s), and flying light helicopters is both difficult/stressful, extremely expensive ($200/hr) and fairly dangerous (motorcycle tier or worse). Those who do it, do it as a hobby. https://philip.greenspun.com/flying/helicopters
Probably because they got rich by being able to recognize a bad idea.
Also, being rich, they probably want to live longer, not die in a half-baked, worst-of-both-worlds technology.