For the cost of living thing, ideally you just implement similar good urban planning across the country. The reason some places are so expensive is because they have relatively livable cities compared to most of the country, so people want to move their. If you just improve the cities in places people already want to be for some reason or another, then you'll just get more people across the country interested in being there unless they have similar options near them. Guess you could alternatively make enough housing for like 50 million people in that one city. Technically, there's always the excuse of "you just didn't build enough". Not sure how the cost per housing unit gets for super structures, particularly since the cost of them includes infrastructure costs we don't usually value into the cost of the home (pipes, roads, etc) and commercial spaces + residential which would make a small city with huge population possible.
I think when a lot of people say high density, they don't mean 100floor residential buildings are all the housing, but tend to think of something more akin to densely packed midrise buildings and green spaces. If you have the later, there simply is not space for cars and high density. Large universities come to mind, where there may be 50k people using 1-2sq mile while 5+ story buildings are rare. You would have to walk a mile or two to get to a car to drive 6 miles around to the other side of campus at 5mph to walk a mile or two to get to you class 1000ft away from your starting point if they were car centric.
You don't even need public transit at that level of density but it's an option.