While I don't want to disagree with you here, because I think we do need higher density housing, and cheaper housing construction methods, along with a modernized approach to urban development -- I would like to point out that there isn't anyone in North America, in the 21st century, that wants to live in a city like Boise. It's a second or third tier destination at best, and it doesn't have the local economic resources to handle sustained regional immigration like this. Very few places do.
That said, I think the real solution here is to be build entirely new cities. It's the last part of your post that got my attention.
Why not build more places that people actually want to live?
This can be accomplished a lot of different ways. There are tax incentives that can be had. Puerto Rico has been growing thanks to its status as an economic opportunity zone. The same thing could be done state side in any number of places.
There's no shortage of land to build cities on, either.
Population density in America is only a problem if you look at the existing cities. If you look at the numbers by square foot, or even square mile, you'll see that there's an absurd amount of undeveloped land in this country that could be developed tomorrow. Taking this into account, overpopulation is only a problem if either new cities are not constructed, or the existing cities we have won't scale (a lot of the latter).
By itself, high rise apartment buildings in existing cities, while a good idea, is only a band aide, which won't solve the larger problem.
We have both the market need and the resources to make a dozen new cities a reality today.
It's not like we haven't done this kind of thing before.
All that's missing is the political will.