> And even if you “win”, the result is a more livable city that attracts even more people, leading to price increases nonetheless.

Your statement boils down to "supply can never outstrip demand" which is trivially falsifiable in its own right. No matter how livable a city becomes, at some point, you can have built more than the housing demanded by people. There are a finite number of humans and the proportion of them who both CAN and WANT to live in these cities is almost certainly manageable.

But don't take my word for it, look at the actuality of Tokyo (which for multiple reasons has a far more difficult challenge than Bozeman or any of these other mountain towns): https://marketurbanismreport.com/blog/tokyos-affordable-hous...