I’d put them near the bottom of the list of “things that make housing expensive.”
At peak, there was about 20-40k listings in greater Los Angeles on Airbnb.
There’s roughly 19 million people in greater LA. Average household size in the US is 2.6 ppl. Let’s round up to 3 and be generous.
That means there’s roughly 6.3 million households in greater LA. Even if 100% of those Airbnb’s are suitable as full time residences (its probably more like 30%), the 20-40k units wouldn’t even cover a single year of newcomers to greater LA (yes, the core is shrinking but the greater metro is growing).
The main (key word) reason housing is expensive is due to geographic and local government restrictions on zoning and development which chokes off supply relative to the number of people who want to live there. Not some silly web app for millennial travelers.