Look at the mess the UK rail system is in - absurdly high ticket prices, confusing interconnections, bad service, finger pointing, bad maintenance, it's a mess.
The original railways were privately owned and apparently worked pretty well but it is a different world now. Minimal costs trump all else and that's no way to run a city's mass transit system. They will never be profitable enterprises, they are a public service, for the good of the public and the city itself.
Ever wonder why airplanes had wifi before the subway? Private vs public.
Ever compare the privately owned NYC ferries (fully open to the public) with the publicly owned ferries?
I'm wondering if ferries are easier to run as there is less infrastructure. There are the equivalent of stations and maintenance depots, but not rails. I don't know who owns the ferry infrastructure or if it is shared.
If you privatised the trains, would the different operators share tracks and stations? If so who would own and maintain them? My guess is the operators would baulk at the idea of maintaining ANY infrastructure and just want to run trains leased from the govt, over a system maintained by the govt, who now has less revenue from the fares.
In NSW our railway barely if never got to operational status under private ownership. The govt took it over immediately and it has been priceless for the state ever since. But of course bean counters have been complaining about it for decades now. I think there has been privatisation stuff going on for a while like the airport line. They were also trying to disentangle the various routes so they could sell them off, but that is hard because they didn't need to be planned that way when built.
It seems to me like a city's metro system is too important to leave in the hands of private operators whose only reason to exist is to make a profit and that will bail out the second things are not worth its time and money. If the metro chokes, so does the city.
Who owns and builds roads and gas stations? Private bus companies, taxis, rideshares, chauffeurs, and shipping companies run on these.
Who owns ports? Private cruise lines, ferries, and shipping companies run ships out of these.
Who owns freight rail and associated stations? Private freight and logistics companies move products over these.
Elsewhere in this thread are comments about Tokyo's private subway companies (three of them).
Airports (and baseball stadiums) are built with public money because of competition between municipalities. They are primarily funded by the people who use them though.
Gas stations are private enterprises.