Go tour some unincorporated land in the Southwest and you'll see how cheaply homes can be built. The problem is that once you're spread out enough to be able to do that, the resources required for daily living have gone up significantly because they're no longer amortized over population density and time. And that same gradient applies to suburbs/ruralish areas with building codes.

Building a bunch of housing in the middle of nowhere for unhoused people isn't going to work unless you also provide them heating, transportation, etc. The least expensive way of doing that is to make it dense like a city. At which point you're just building a planned city, which is fine, but not particularly in the purview of the US.