I found that cold sendings never seem to work for me. The bway shows I've PAed I have all gotten through recommendations or by knowing someone on the SM team. I still had to interview, but I find the PSM feels more comfortable hiring knowing that someone they trust has recommended you. I will also echo that I have yet to meet a non-AEA PA on broadway, they are all, like myself, young eq-SMs who are trying to get their foot in the door. Also, since it is very common the the PA will become the sub once the show is up, most PSMs want the PA to be eq.
If you are dead set on nyc, try looking at companies that have the different levels (ie -MTC or Roundabout). Since they have different spaces it is possible to PA, say at City Center (with MTC) or the Laura Pels (with Roundabout) and then move up from there. There are also tons of off-broadway theatres that might be easier to get into as a non-eq PA.
Also, summerstock is a great way to make connections, a lot of ny SMs go outside the city limits to stage manage during the summer. There are tons of internships and PA positions out there where the contacts you meet can be extremely helpful. Places like Williamstown, Hangar, Goodspeed, most Shakespeare Fests, etc, etc. (this list can go on) have well known directors and SMs who are nyc based that become great contacts when you return to the city.