No, we have been ignoring your point because it’s one of those Econ 101 world views based on assumptions that have no basis in the real world and actual human behavior. You can’t look at the number of Airbnb rentals and conclude it’s not having a significant impact on the market simply based on the raw number of units. What we know from all systems theory is that a small portion of things has the majority of the effect. In this case, short term rentals are very likely setting the upper bounds of the pricing range.
"At the median owner-occupancy rate zip code, we find that a 1% increase in Airbnb listings leads to a 0.018% increase in rents and a 0.026% increase in house prices."
Clicks on "actual research," finds the usual econometric voodoo. Even better, the use of instrumental variables to supposedly "isolate" effects. Totally representative, I am sure.
PS: the author's own words contradict your claim that the effect is tiny, although again I think this is a serious undercount resulting from a poor model and ever worse availability of data nationwide:
"This means that, in aggregate, the growth in home-sharing through Airbnb contributes to about one-fifth of the average annual increase in U.S. rents and about one-seventh of the average annual increase in U.S. housing prices."
https://hbr.org/2019/04/research-when-airbnb-listings-in-a-c...
https://time.com/6223185/airbnbs-empty-short-term-rentals/