AirBNB could not exist and all our cities would still have severe housing shortages. It's a scapegoat, the real problem is lack of supply, which is a zoning regulatory issue.
Zoning prevents building the stock needed (or things like making styles of homes like multi-family illegal), so AirBNB may be an irritation to it, but its far from the root problem.
https://www.dataforprogress.org/housing/
> Without abolishing apartment bans we are left with very little space in growing cities to place social housing. Let’s say we raise taxes on the rich as much as we dream, and set about to build social housing apartments. Right now, apartments – market rate or not – are illegal in much of the useable land in American cities (only 17% of Seattle’s buildable land allows apartments, for instance). Where will we put the new social housing if apartment bans remain? (And it will take years to build the social housing we need; in the meantime let’s at least build some places for middle class and working class people to live.)