First, in most cities, it is far too difficult to bring new housing units to market due to slow-moving non-deterministic regulation. Replacing that with efficient and deterministic regulation is the most important thing, so that over time the market can supply enough housing to bring prices down.
Second, in strained housing markets (most major cities), property taxes need to go up significantly to reduce the utility of housing as an investment, and there especially needs to be a substantial tax penalty on vacant and semi-vacant (second, third homes etc.). There’s no reason a housing strained city shouldn’t use that mechanism to raise funds from the wealthy and discourage vacancy.
Finally, cities that are sufficiently strained should impose significant hotel and vacation rental taxes. That’s harder to implement ordinarily because it’s not actually that easy for the government to tell the difference between a normal rental, short-term rental, and vacation rental. Also, short-term rentals are actually a healthy part of a local economy. But assuming that they started by making it much easier to build new supply and reduce the housing shortage, in the short term the city should be profiting wildly from tourists rather than letting tourists displace local residents. That tips the economic balance by lowering the demand for tourism to levels that don’t distort the local housing market.
Over time that should be able to bring housing supply and demand to an equilibrium that is affordable to the locals, and at that point taxes could be adjusted over time (likely reduced) to maintain the equilibrium.
Sadly, most people who work in City government have a really poor understanding of economics. They lean utopian socialist and are really offended by things like low income housing which tend to be ugly, and so they strangely and unintentionally ally with the hyper wealthy to try and exclude anything that offends their aesthetic (ie the poor), then fail at providing state housing that has the aesthetics they prefer because it’s economically infeasible to provide luxury housing for all. I wish this would change, but I fear that things will have to get much worse before this deeply entrenched dysfunction can be rooted out.