A big part of the problem in Manhattan is that most places don't have alleys. There is nowhere for the trash to be left except on the sidewalk.
Some cities have the trash trucks drive around playing music, ice cream truck style, and you are required to bring your trash out to the truck. Logistically that would probably work in Manhattan. Politically I don't think you would ever be able to get it done.
Most places in the US don’t have alleys. The problem isn’t the alleys, it’s the trash cans. Most modern US cities have cans that can be lifted by the truck, and then emptied. NYC, on the other hand, insists on just leaving everything in flimsy bags, open to the rats.
Most places don't have a mostly continuous street wall with little/no setback from the public street/sidewalk.
Part of the problem in Manhattan has been that many buildings really do not have any place they could store those sorts of bins, or at least not in the quantity of them that they need for their trash output.
Manhattan isn't unique — and as far as I've heard, they've recently copied systems in use in Madrid (bins in parking spaces) and London (wheelie bins) to solve the problem.
Some cities have the trash trucks drive around playing music, ice cream truck style, and you are required to bring your trash out to the truck. Logistically that would probably work in Manhattan. Politically I don't think you would ever be able to get it done.