In its glory days (the days when dozens of new stations were opened every year) it was private companies competing for customers.

Imo that's not what happened.

This runs entirely counter to the whole design of the subway system.

Before the subway, everyone was crowded into lower Manhattan. Rich people lived uptown, but the outer boroughs (with the exception of what is now Downtown Brooklyn) were fields and forests, almost completely undeveloped.

With consolidation, the city had tons of land available for people to develop in the outer boroughs, but because commuting to and from them was impratical, developers didn't want to build anything.

Enter the subway. Not only are 4/5 boroughs tied together with a rapid transit system, but the fare is kept artificially low in order to allow working class people to move out of Manhattan. They can afford to live in Brooklyn or the Bronx in part because they only have to pay a nickel to get to work.

The only railroads in NYC capable of operating the subway were in no way willing to build it. Too much capital, too much risk, too little foreseeable benefit. It's almost impossible to make transit profitable, even back then.

So the city provided the capital and recruited the engineering talent to build the system, then allowed the private operators to run the system after construction was finished so long as they kept the fare to 5 cents in perpetuity.

Of course, neither the BMT nor the IRT could possibly operate profitably like that, and when the city intentionally cannibalized their ridership by building the IND, they folded, the city consolidated the three operators, and NYCT was born.

You can't operate the subway profitably. The whole purpose of its existence is as a subsidy for employers, particulary in Manhattan. Any private carrier would have to increase the fare or cut service and that would defeat the whole point.