Cars can be a psuedo-free market, because you can buy a car built in Detroit, but drive it in Boston, and everything is fine. You can buy a car from Tokyo, and drive that in Boston and everything is fine. You can also choose
not to buy a car, but instead buy a motorcycle, and everything is (mostly) fine.
You can't do that with housing. No amount of new zoning or new construction can solve that problem, because the land in every city is already fully monopolized.
Housing will never ever be a "competitive" "free" market. It can't be, because buyers don't have meaningful agency in their choices (for all but a few exceptions, everyone basically has to live pretty near their work/school/family), and buyers can't opt-out of the housing market (there are no reasonable alternatives except things like homelessness, which aren't reasonable).