Subway systems are a natural monopoly; privatising something when there's no real competition has limited benefits and can be actively harmful if it reduces accountability. It works in Japan because under the keiratsu system nominally private companies are still closely tied to the political system and each other, so they have accountability through that channel.

Also a lot of the value created by a subway shows in land values rather than being captured directly in fares - the article says the MTA is 60% funded by fares which is high but, well, not 100%. You need a mechanism for those who benefit (i.e. owners of buildings near subway stations) to fund the subways. In some countries the transport authority will essentially own large shopping mall complexes around each station and fund the subway that way; in Japan employers often pay for workers' subway tickets.