This is true. The rich would like to relegate the poors to public transportation to keep the roads clear and keep undesirables out of their neighborhoods.
The "that's an advantage for the rich" has an easy fix: ban all city traffic except commercial vehicles for delivery of goods, ambulances, etc.
A good public transport is a boon to the poor and even middle class. In Europe, in cities where it's available, even the rich (the top 10%-5%, not the top 0.01%) routinely take it.
We do need to do something about the "holiday conundrum", for lack of a better name. That it, there are times during the year, most notably around Christmas, where suddenly everyone in the city gets time off and wants to travel to visit their family out in the middle of nowhere - and that also includes the operators of public and commercial transportation networks. You have a sudden influx of commuters at the same time as transport service capacity drops to a fraction of the normal.
Not sure how to handle this short of abolishing holidays entirely
(Not necessarily a bad idea - too many things in society already run in synch where they shouldn't; see e.g. most people working 9-5, including services all those people need, so there's e.g. no good time to go to a dentist or visit a bank without taking a day off at work.)
There's a saying to the effect that a healthy society is not one where poor people can afford cars, but one where rich people take mass transit willingly.
Yup. And a big part of it that a lot of HNers don't seem to want to hear is that to get rich people to want to take mass transit, you need to strictly enforce against anti-social behavior. Treating the subway as a shelter or being menacing or foul-smelling should get you swiftly ejected.