Wouldn't this mean people who earn enough to rent, but not buy (ex. service workers) wouldn't be able to live there (since there wouldn't be a stock of rental housing they can rent), and thus would have to commute?
Why wouldn't there be a stock of rental housing? Nothing stops a full-time resident from owning rental property. If you want to own one (or more) houses there you have to live there. Seems perfectly reasonable to me. I wish this could be tried in the USA, but voters (rich property owners) would never allow it. There are a lot of places (not just ski towns) that are ruined by rich people bidding up property and then leaving it vacant 358 days of the year.
I don't know about Tirol, but in Banff, the restrictions don't prevent construction of multi-unit buildings specifically for renters. I don't know exactly how ownership of these buildings work, but I think they're owned by the town through some non-profit org.
I don't live in Banff and I'm sure there are issues with this system, but it seems pretty reasonable. Service workers would definitely be priced out if every condo/apartment was owned privately and available as a short-term rental.