My personal feeling is that you naturally learn how to deal with the very unique situation that is riding a bike in NYC without dying or causing bodily harm to yourself and others.
Personal example: One afternoon, I happened upon a poor guy on the Brooklyn Bridge (the cement part) who had just done an endo and smashed his face open. This taught me several valuable lessons, 1) Always wear a helmet, 2) speeding on the Brooklyn Bridge crowded with tourists means certain doom, 3) learn how to brake without flipping over.
There are plenty of other examples, such as almost being doored, nearly getting run over, dodging oblivious pedestrians (a creature that I am guilty of being from time to time), etc.
So I think saddle time is the most effective way of becoming a safe rider. Unfortunately because of the large numbers of new riders, we'll be seeing a lot of sidewalk riding, salmoning, riding without lights at night, and other generally dumb things for a while. The good news is that once this awkward stage is over, we'll be in bike utopia! Right? Right??