The real problem is putting operations and verification in the same hat - public or private. Private transit systems work well when government regulators have the power to set and enforce a strict SLA. They fail without oversight. Same with public systems. When the operational and regulatory agency are the same, it becomes hard to penalize failures.
Tokyo Metro is owned by... the national and local government.
From Wikipedia: [1]
The Japan Railways Group, more commonly known as JR Group (JRグループ Jeiāru Gurūpu), consists of seven for-profit companies that took over most of the assets and operations of the government-owned Japanese National Railways on April 1, 1987. Most of the liability of the JNR was assumed by the JNR Settlement Corporation.
Granted, that not every experience is abyssimal, but in my experience most train riding experiences in the UK are pretty terrible.
Filthy trains with partially massive undercapacity at sky high prices. Delays, cancelled trains and you - the passenger - is just left out to dry.
Infrastructure is also problematic, like an underabundance of ticket machines, let alone ticket selling staff at stations, etc.
It's also interesting to note that Japan rail companies can keep prices reasonable by making major parts of their income via real estate.
Which New York system? The New York MTA is the result of consolidating multiple, unprofitable systems (e.g. LIRR, New Haven, Staten Island). Most were snatched out of the jaws of bankruptcy by the government as commuter service just isn't profitable much of the time. Let's not forget that the private rapid transit (BMT and IRT) didn't work out so well either (bankruptcy and poorly thought out contracts) and eventually was acquired by New York city.
That is not to say you can't have private systems that work (Tokyo, Hong Kong). New York just isn't a particularly good example.