I’m confused by what claim you’re trying to make here. Are you suggesting that low income people are going to move into rich neighborhoods to be closer to work if we eliminate public transit? Public transit enables higher density, which increases the visibility of low income folk by pushing rich and poor people closer together. It has the exact opposite effect you are suggesting.
>Public transit enables higher density, (...) pushing rich and poor people closer together.
The later doesn't follow from the former - there's nothing about public transport that inherently clusters the affluent around it. Good schools & "nice" neighborhoods would be an overriding concern.
Private transport enables[1] mixing and matching, by the virtue of being rather flexible and quick to react to changes in supply & demand. There's nothing inherent in private transport that would cause self-segregation, nor that would allow planned segregation. Your point about density holds true, but it's still a low-density somewhat homogeneous society.
Public transport, by the vice of being mostly fixed in place and centrally managed, is how one channels people into pre-defined paths and spaces. Sure, higher density is possible, but enclaves of the reliant are created around the transportation hubs, while the affluent can settle anywhere - presumably with no unseemly bus stops & metro rails in view.
Build a bus or metro rail stop, who do you think will tend to settle around it?
Have you ever been to a an actual city in your life? The most affluent neighborhoods are the ones with excellent transit. Look at downtown and midtown Manhattan.
