OOC:
I'll need to bring in some data to this because the discussion is already going off the rails.
Historically, what became the New York Subway system was first planned after the blizzard of 1888 effectively shut the city down. Ironically they have now started to close the subways during blizzards and major storms. Construction started in the 1890s.
The Chicago El predates the New York subway system, though the Loop was only incorporated, with difficulty, in the 1890s. The oldest subway tunnel in the USA is in Boston, but the T is slightly younger than the New York subway, which was originally supposed to be an El on the Chicago model.
This is a minor point, but there were some pre-existing railways to bring holiday goers to Coney Island that got incorporated into the subway system. There were also lots of streetcar lines, later replaced by bus lines, so its not like there would have been no mass transit without the subways.
Anyway, construction on the system started in the 1890s, I think in 1894. Greater New York was incorporated in 1898. Before 1898, "New York City" referred to Manhattan, though the Bronx was added in 1895.
The Census had the population of Manhattan at 1.4 million people in 1890, increasing to 1.8 million in 1900. As of 2010 the population was 1.6 million, for comparison. The New York City population post consolidation was 3.4 million, as opposed to 8.2 million in 2010. So in 1900 1.8 million people lived in Manhattan, another 1.6 million in the outer boroughs. The outer boroughs effectively absorbed the net population growth in the twentieth century.
Here is the point. The subway system is not required for New York to be a major city. New York was already one of the most important cities in the world, and the largest in the United States, before any part of the system had been built. Manhattan actually pretty much reached its present day population level before the subways. And without the subways you would still get mass transit in the form of streetcars and busses. What the subways seem to have facilitated was people living outside of Manhattan and commuting to jobs in two centralized business districts on the islands, along with the PATH system and the LIRR, MetroNorth, and New Jersey Transit commuter rail.