Have you spent much time in Tokyo, Seoul, Hong Kong, Singapore, Shanghai, or Taipei? In all of these cities, public transit doesn't just "have its uses," its the primary way of getting around the city for the vast majority of residents. This isn't about social policy. These cities span the entire range of social policies.
All it takes is a government dedicated to building a comprehensive, functional, and intelligently designed transit system. There are still crazy people on the subway in Shanghai. But because its the best way to get around the city, everyone takes the subway. So maybe you see a beggar or a homeless person every couple of days, much like you would when you're out walking around.
The problem we get in most American cities is that the transit systems are terrible, so anyone who can afford to drive takes their car, and transit systems end up existing only for people who don't have any other options.
So you decide to take the bus one day, and everyone else on the bus is either homeless or crazy, and the bus has to take some ridiculous route to pick up enough people to justify its continued existence. So you never take the bus again, because it was unpleasant and wasted your time. Now we've gotten into a vicious cycle where the awfulness of the system ensures that it will continue to get ever worse.
This was a long rant, but what I'm trying to say is that public transit isn't inevitably broken, as many of the world's most successful cities demonstrate. It just can't be half-assed, and it feels like we've half-assed it in most of the US.