Ignoring the political backlash around taking people's parking, I would instead point out the likely rate of hitting dumpsters in the street with how tightly packed people would park around them.
NYC was, for large stretches, built without the alleys that lots of other cities would use to store such bins, and I don't think you can easily accommodate the volumes of trash that people want to turn over with just street storage without significant logistical problems with how much you disrupt the flow of people in the city.
Even if we eat the cost of parking around residential locations, businesses have gotten used to using the space in front of their restaurants as additional seating, so cannibalizing that for dumpsters will cause massive backlash on that front, to say nothing of the nasty effects on people's desire to eat when sitting next to a dumpster.
I agree that NYC is slow moving on a lot of things, but this is also a genuinely hard problem.
