This is such a fundamental misunderstanding of how the housing market works, and totally ignores apartments/midrises/highrises. The problem with housing in the US lies with overregulation (mostly in the form of zoning). I know this well, as someone who has attended numerous planning meetings in Chicago, I have seen many buildings reduced in unit count solely due to NIMBYs (mostly retired people, as these meetings are often during the day or the immediately 5-7pm time when many people are still commuting home from work).

No idea about the US, but that is indeed how it happens on this side of the pond - developers rarely build mid or high rises, it's all single family homes.