If we built lots more housing, there are two mutually exclusive possibilities (though honestly, they are on a spectrum):
A) Housing becomes affordable, or B) housing stays expensive.
The latter means we added trillions of dollars to the national wealth.
By the way, please go ahead and do tax property a lot. Ideally, you'd only tax the unterlying land with a land value tax. But even a naive property tax is OK.
That way foreigners buying up local real estate for speculation means foreigners paying for local services.