You haven't seen "overrun" or "bursting at the seams" anywhere in the US.
London charges a fee to drive within the central city. Perhaps that would help congestion in NYC as well.
I found the subway system to be quite nice when I visited NYC a couple of years ago. I would certainly use that myself. I don't know that I would have the courage to ride a bike in NYC traffic. And the weather in summer or winter would be no joy for bike commuting, either. People simply dress too well in NYC to put up with the mess you will look like riding a bike. And the thing that really attracts you to NYC is all the well-dressed people out walking. Riding a bike would separate you too much from that in my opinion.