Working backwards:

Reducing immigration would also reduce the tax base, reducing income, reducing benefits, etc. (check out this story).

For homes, you glossed over the millions of Millennials that stayed in apartments for much longer than previous generations.

We're not talking about "housing units", which includes apartments and whatnot. We're talking about single family homes that people can outright purchase.

Hell, even apartment space is at a premium, and we're seeing the increase in rents to show for it.