"New Yorkers avoid the subway because it is essentially a mobile penitentiary and freak show." Really? And when was the last time you rode the subway? Because I think you have some rather unusual ideas about what it's like not based in reality. New Yorkers avoid the subway because it's crowded, and especially because it's slow. Line speeds are something like 12 mph for local trains, and when you add in the walk to and from the station, the waiting for the train, and the chance of "unavoidable delays".

Anyway, you're quite right that cars are the biggest limiting factors in how many cars can fit in Manhattan. The way to solve the traffic problem in that framework is simply to keep reducing space for cars, and increasing it for other uses such as pedestrians, bikes, buses, and streetcars. The less space cars have, the less space there will be in which to create gridlock, and the less gridlock there will be. But what if you don't want gridlock? Then you make it so that there is some other limiting factor, for example the expense of driving, via some mechanism such as tolls, congestion pricing, parking pricing, and so on.

To paraphrase Yogi Berra: Nobody rides the subway, it's too crowded.