I think you are more right than you realize. The US Federal government still to this day subsidizes and promotes home ownership. The mortgage tax deduction, the Home Sales Tax exclusion, and even the fixed rate 30 year mortgage itself - without which most of us could never buy a home in the first place - subsidizes home ownership.
But the actual building of homes is left to the private sector, which is frequently banned from building new homes by local governments.
It’s a strange situation: the Feds promote home ownership, while local governments make home ownership difficult by limiting the number of homes that can be built. It’s little wonder the housing market is so dysfunctional.
That might be technically correct on the build part, but the US government used to subsidize home ownership through loans as described in [1], e.g.
> GI Bill benefits that guarantied home loans helped many Americans buy houses and move into the suburbs.
[1] https://www.exploros.com/summary/The-Growth-of-Suburbia