Hypothetically: I have X dollars in my pocket for a trip to NYC. I can spend it on my hotel + the taxes that the hotel pays, or to an AirBNB host. All things being equal, let's assume AirBNB is cheaper for me because hosts don't pay these taxes, which means I get to keep a larger % of my X.
If I go for the hotel, I'm paying a larger % of my X for the rental (some of which is allocated to taxes), and less stays in my pocket, which means that less will be available for me to spend on the local economy (though it should be benefiting us locals indirectly via taxes, or so goes the theory.)
I'm not going to somehow magically find more money to spend in NYC if I choose to go with the hotel - I'm just going to spend less money, because I have less money, because I've spent more on my hotel.
Your math assumes that somehow I have the exact same amount of money to spend whether or not I choose AirBNB or a hotel, and also that the same number of short-term tenants will visit NYC regardless of the cost they realize paying for AirBNB vs. a hotel - and that further that these short-term tenants will spend the same amount of money on goods and services regardless of their rental costs.
I'm going to go out on a limb and say that's unlikely. :)