Public housing does help address one of the roots of the problem - this is a problem with multiple roots. Public housing does not need to factor in rent seeking in their pricing, as housing is end itself, not the means to another end. It also helps address the other factor of financing housing construction at scale - build standardized and in bulk, and you bring down the price of construction, which is not insignificant.

The government is also in charge of setting zoning policy, meaning that they are better positioned to change said policy.