Generally, self driving cars are very safe, too safe actually. There are some high profile accidents, most of them involving Tesla because of the scale and way they approach the problem, plus an infamous one by Uber, but generally, self driving cars will not go anywhere unless they are sure the way is clear.

And this is the problem, self driving cars don't force their way, and they break when anything is vaguely in a collision course. If there is a problem putting them in NYC traffic is that they will stop all the time and create traffic jams, maybe get rear ended by other (human) drivers and fail to get their passengers to the destination because of some minor obstruction.