The high rent is deliberate policy. You are kinda missing the point. The reason NYC is safe now whereas it was a war-zone in the 80s is all the criminals had to move away because of higher rent. The government deliberately got rid of most of the section 8 housing and constricted the supply of cheap apartments. They want to price people out.
There are still a lot of housing projects throughout Manhattan, sometimes adjacent to very expensive housing. An example is the largest projects in Manhattan, Baruch Houses, 17 buildings, 27 acres. This is right across the street from The Ludlow which most recently rented a one-bedroom for $4,485 a month.
