Tight housing didn't solve that. In fact, everywhere there's a homeless camp happens to be right where there is tight housing. Because the corporations building your dream community of prison block housing is still charging a lot for it. It's not creating more housing for the poor. That's just what the developers' marketing agency told you. Housing shortages have nothing to do with land shortages. They have everything to do with banking and mental healthcare.
If you have an engineering solution that can build skyscrapers on begging income, land prices aren't going to hold you back from being a billionaire...
Specifically for housing in America, though, more houses don't lower the price. That's because the more housing is owned by an individual builder the more they control appraisal comps. Why do you think they have so much marketing to try to convince you that it's good for the environment?
They are able to get zoning exceptions to build high density, high priced living. Pick any skyscraper residence in America. Nearly all, if not all, are far more expensive than surrounding suburbs.