The problem is that people, unless they can afford to buy, are forced into renting from a private landlord and working to pay off someone else's mortgage. If you have enough livable public housing such that anybody who needs a place to live can get a place on the public system, the power balance changes and private landlords aren't so much of a problem.
The solution to a broken market, is to fix the market. Write the rules / laws to encourage competition and thus the desired outcome. If something is a natural monopoly then make it a utility, regulate it properly, and have the competition at the worker union and/or contracted workers layers.
The only way to fix too expensive housing is to build enough housing to drive the cost down via supply. If there's enough 'close enough' to where people want / need to live, then if someone asks too much it won't sell. Taxes that incentivize one (or at most two for a summer/winter home) dwelling per person and the 'Land Value Tax' for density and under-utilization issues help prevent unwanted speculation and rent seeking in the market.