What happens is that a certain, usually small, percentage of residential apartments is made available to the tourists.
Each apartment means about one/two less hotel rooms.
Hotel rooms take space too, they are not floating over the city. They often replace old residential buildings to build hotels effectively taking away space from residents who are pushed out of the city.
So we may say that tourism takes space in the center, not apartments.
I didn't hear much complain about this neither, as tourism brings money and, if anything, they want more of them. It seems to me that NY has a very specific situation so Airbnb may be actually harming the residents, I don't know, but in general Airbnb is not doing anything particularly new in Europe, where apartments for short term rent have been existing for long time.
The hype is all about "rent from another human being" but the reality, and I guess most of their business in Europe, is simply providing a better platform.
Maybe NY regulators could look at Prague or Croatia to solve the problem.
[edit: formatting]