dense housing is frequently not permitted by zoning. sometimes housing itself isn't permitted.
> What is left is to build more dense (replace single family housing with apartments or town homes).
yeah, that's exactly the thing that's banned in very many places where it would be most useful.
But where they are permitted, housing prices are still high. Higher even, like NYC and much of New Jersey near NYC. Is there a good example where increased density has led to lower prices, not higher ones?
Second, the economics of building dictate that skyscraper-level density will only happen in places where housing is relatively expensive. But there are plenty of examples of density leading to lower prices, in the suburbs.
Do you have specific examples? The only places I can think of are economically depressed (e.g. in the midwest, or in Florida swampland) and aren't particularly dense.
Maybe Houston? Houston is famous for having almost no zoning, but even Houston seems to be heating up these days.