That's because very few people want to live in the 80% of Nevada that's mountainous rugged terrain or sandblasted desert, free of water, power, jobs, or infrastructure. Let's fix the problem in the places where people want to live first before we look at the federal government.
The federal government owns land that nobody wants to build housing on. It's not enough just to have a house; housing is only viable in places where there is an economy to support the people who will live in the house.
You wrote about what "we" need without identifying whom else that statement represents. You may prefer higher-density housing. It may even be the case that a majority of people near where you live prefer high-density housing. This is not a universal preference nor objectively better. Those who do not share your particular taste do not necessarily prefer "living super far away from others" either.
Let's begin by acknowledging the vast middle ground between the two extreme positions and also downsides of population density.
What does utopia look like for mice? According to a researcher who did most of his work in the 1950s through1970s, it might include limitless food (of course!), multiple levels and secluded little rodent condos. These were all part of John Calhoun’s experiments to study the effects of population density on behavior. But what looked like rat utopias and mouse paradises at first quickly spiraled into out-of-control overcrowding, eventual population collapse and seemingly sinister behavior patterns.
Can we agree that at least an acre of federally owned land is neither in the desert nor "50 miles from anywhere" and may be desirable for commercial or residential use?
So are you proposing building houses on national forest? National parks?
The vast majority of federal land is BLM, which by its very nature is not desirable for development.
I could go camp on BLM land today in eastern Colorado, it just wouldn't be very nice, because it's scrub brush all the way to Kansas City (exaggerating, only slightly).
Go camp for a month on BLM land somewhere and see if it's a place you'd like to build a town.
Otherwise you're proposing building on protected public lands which I for one would prefer to remain natural.
You can still homestead parts of Alaska. If you can develop the land it's yours to keep, and I do imagine that the feds let go BLM land on occasion, but it's not going to solve housing as suggested up-thread.
Something even more rarely mentioned is that urban and built up areas are just 1% of habitable earth, forests (37%), urban crops 10% (providing 82% of calories) and animal agriculture 35% (for those remaining 18% of calories).
If we'd switch to plant based diets, we would globally free an area the space of both Americas. If this area were reforested/rewilded, we'd store enough carbon to stop climate change, stop biodiversity loss, droughts, soil erosion, eutrophication and anthropogenic oceanic dead zones, and we'd still have plenty of space to build housing for everybody.
Much of the land in the US isn’t suited for residential or commercial purposes. Building in some parts will just mean expensive water infrastructure demands and/or fire risk. That being said, there’s large swaths of Vermont, New York, PA, and the southeast which do not have these problems.
No, I prefer National Forests that are open to my recreation than barbed wire and no trespass signs.