I'd also venture to guess that in many places where Airbnb is popular (New York and San Francisco, for example), most of the company's partners are tenants, not homeowners. That has a whole bunch of added complications, but it pretty much throws the whole "private property" argument out the window.
How does a tenant subletting space diminish the property rights of anyone? While subletting isn’t as common as it once was, it still does exist. It’s both legal and moral. All a tenant is doing is assigning all or a portion of their property right, their tenancy, to another person.
As for zoning, there is no place in local governance that is more rife with corruption than zoning. There are places in the US that don’t have zoning such as Houston. They’ve managed to survive. Deed restrictions prevent rendering plants from being put next to daycare centers.