It's not necessarily only window units. Many apartments have in-wall units also. It would be cost prohibitive for most to close up a hole in a brick wall to install a split unit.
I guess this might be an example where leapfrogging technology by being a bit late ends up with better results? One reason split units are popular in southern Europe is precisely because they're cheaper to install than wall units, since you only have to drill a small hole through the wall, not open up a big gap. But I can see that if you already opened up a big gap, because you installed A/C back when the ductless split units weren't widely available, then it'd be more expensive to close it back up again. A/C has only really gotten common in Italy, Greece, and Spain since the '90s or so, so they went straight to the modern split units when retrofitting.
This doesn't explain NYC, where a huge number of apartments use window units. Landlords don't feel compelled to provide A/C, so tenants need to provide their own, but tenants aren't allowed to drill any holes at all, so they're left with window units. Plus, window units are cheap as chips in NYC--under $200.
In any case, almost nobody in NYC uses the sort of split system so common everywhere else. Older buildings are left with window units, while new buildings have built-in units, often with the most horrible control panels you can imagine from some dudes in New Jersey.
My building, from the 80s, has no central air, but through wall cutouts the size of a window unit. However, when I went to buy an air conditioner I found that only one company made a unit the exact right size and the cost 3x more than the BTU equivalent window unit.
How about using the radiators and district cooling? Distributed hardware is usually not very efficient, especially here with space and noise constraints.
Radiators do not work for cooling. Cooling creates condensation, which will drip on and rot the floor unless you move it out of the building somehow. Cool air also needs to be forced to circulate, whereas hot air will do that on its own.