Housing isn't the problem here but as you said correctly, it's the land ownership that is problematic. If we assume that house owners merely rent the land then it's perfectly fine to build a private housing market since housing needs can differ drastically between citizens. Some need a big house for a large family with lots of children, others only need a studio apartment. Some people want a home workshop or a home office. The market finds a satisfying solution for everyone.
Land ownership is problematic because neighborhood improvements can be extracted via land ownership. The most effective policy against this extraction is to simply tax the land itself and thus turn it into a liability if the value of the land grows. Ideally, property taxes would be replaced with land value taxes.
If land is a liability the only reason why people would purchase it is to improve the value of the land and make it available for productive uses like renting it out to a large number of people.
Land ownership is problematic because neighborhood improvements can be extracted via land ownership. The most effective policy against this extraction is to simply tax the land itself and thus turn it into a liability if the value of the land grows. Ideally, property taxes would be replaced with land value taxes.
If land is a liability the only reason why people would purchase it is to improve the value of the land and make it available for productive uses like renting it out to a large number of people.