> The only solution to the underlying problem is to allow construction to meet the underlying demand.
While I do think that allowing more construction would be great, there's probably other things that could help alleviate the problem.
1. Limit the number of units that can be rented by a single landlord in an area
2. Limit the proportion of units in an area which can be rented (maybe creates a sort of taxi-shield problem which might be it's own weird thing, but that's a problem for multi-property landowners, not single family homeowners).
3. (as OP suggests) limit the number of nights/year which can be rented to some number which would be very reasonable for a resident-owner who's out of town sometimes but very unreasonable for owner-investors.
4. Maybe some kind of hybrid approach - you can rent out space in your primary residence all you want, but your nth residence is only rentable 1/f(n)% of the time (for some reasonable f(n))
All of the above are presumably best implemented by governments rather than a well-meaning airbnb (who, even if they existed could presumably be overpowered by a less-well-meaning competitor)