Regulation around short term rentals and regulation around construction and zoning are two very different things. I agree that every city can probably benefit from more progressive construction policy, but that doesn't mean that restricting short term rentals isn't also a good idea.
It's nice to think that the free market combined with liberal policies will produce so much capacity that all problems around pricing (and by extension AirBnB) will be solved, but we haven't seen much evidence that this can occur in a timely enough manner to come anywhere close to meeting demand, which may also be increasing over that time. Here in San Francisco for example, we've seen a massive construction boom over the last few years, but it's having no effect on prices because even though thousands of new units are coming online, the pent up demand still vastly exceeds the new supply.
Even assuming you could solve the problem with construction, the solution will come on a decade scale whereas AirBnB can entire change the face of a city's rental market in a few years, or less (take a Berlin for example). Popular cities would still need restrictions on AirBnBs to keep the problem in check until longer term increases in housing capacity can be put in place.
And, regardless of the impact on housing prices, it is certainly within the role of civic government to determine that they reject short term renters from imposing externalized costs on neighbors.
(And yes, more cities need to get out of the way of housing being built. Would that other cities had half the construction that one sees in Toronto)
In Toronto they estimate you need to make ~130k / yr to afford a home (any home). Bear in mind that a software dev in Toronto makes ~80k / yr. Up until very recently, our lower wages matched a lower cost of living in the city, but this has been blown up by things like foreign investment, and locals taking out a line of credit on their main home to put a down payment on income generating properties. No other sector in Toronto has seen price inflation like this, it's only been housing.
The problem with the deregulation thesis is that you now have to compete with a lot more people for resources. In the housing case, you now are competing with people who live in completely different economies or have substantial economic advantage over you. As an example, a Toronto home is extremely inexpensive to a bay area software dev making ~150k / yr (CAD ~200k). As a side hustle, this might seem like a good investment, and locals simply can't compete.
(I realise bay area rent is insane and this isn't likely fair without taking bay area cost of living into account, I just wanted to use a relevant example).
My experience sees an average of 10 to 30k less than that within the city save a couple of outlets (at the developer level).
The rest of what you're saying is totally valid. Most of the places I've rented have been owned by people living at (put mildly) great distances from the city. This seems to be a common experience.
I have been on a HOA where we essentially forced an owner to sell their unit as we wouldn't authorize them to rent it out as we were at capacity of homes rented out. The decision was made to not make an exception as once you make one you have very little ground to refuse others.
That said, I do think it is fair for airbnb rentals to be subject to the same taxes and regulations as hotels.
Edit: I don't mean that housing liquidity raises or increases home prices. I think that would be tough to say. I just mean that it's good for the housing market from a consumer standpoint: it creates a kind of flexible vehicle for dealing with short term housing needs. It's better as a society for us to have a way to rent somewhere for a week than to not have such a way to do that (and hotels only offer a very specific kind of residence, which is not to everyone's taste).