At least in LA, the problem isn't "poor people ride public transportation." It's straight up "I don't feel safe." I tried using the Metro and didn't buy a car for nearly a year after I moved here, but there were incidents I experienced that made me abandon it. It had nothing to do with poor people.
"A survey of riders showed that women feel less safe than men during all stages of their transit trips — especially at night. For example, just 13% of women surveyed said they felt safe waiting for a train or bus after dark, compared to 30% of men." [1]
When 87% of women and 70% of men don't feel safe on public transportation that's a real problem and it's no surprise that people abandon it as soon as the possibly can.
[1] https://laist.com/2019/10/03/los-angeles-la-metro-women-publ...
It's not a requirement that these public services break even. In fact, (controversially) it's the opposite. [2]
[1] https://about.usps.com/newsroom/national-releases/2019/1114-... [2] https://en.wikipedia.org/wiki/Deficit_spending
Because people who don't use it don't want to pay for it. There is a lot of opposition to public transport that doesn't pay for itself. There is also the horrible feedback loop of "only poor people take public transportation" stigma, making it more difficult to break even.