Wouldn't private planes still necessitate air traffic controllers? I remember when I flew with my uncle in his small private plane [1] he had to fairly frequently chatter with the air traffic people over the radio to report his position and altitude and the like.
[1] Just a tiny two seat thing, I don't come from billionaires, it wasn't a cool private jet.
No one rich enough flying what the average person would consider a "private jet" or private plane would be flying VFR from uncontrolled airport to uncontrolled airport. The "ultra rich" are not puttering around in single-engine Cessnas