Much of Japan's train network is privately operated. Japan has some of the best transportation in the world.
Take a look at Brightline. Brightline from Orlando to Miami had 2.7 million riders last year. They're already working on Brightline West from LA to Las Vegas.
I think public transportation infrastructure is great for rural areas. It's similar to USPS serving everyone. But if USPS was the only mail carrier everywhere, package delivery service would be demonstrably worse.
What is wrong with both private and public transportation infrastructure?
i think the US lacks the regulatory structure and social character that's more present in Japan that make private-public services more successful there.
as a regular metro commuter, i don't think i'd be totally opposed to private transit in LA if it were heavily regulated. but without that, i'd rather deal with all the problems on the metro (stinky riders, drivers switching mid-route, track traffic) at 1.75 per ride, than any of my money go to making Uber shareholders (or anyone who profits by exploiting the "gig economy") more money.