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.
American government is built on the idea that government is intrinsically both malign and incompetent. That's why they have broken it up into infinite pieces - so that every section of government is constrained by every other section, and then they complain that no section of the government does anything. Why can't the city fix public transit? Well, maybe you should ask the separately funded and elected Transit Authority. Oh, they say there's no point running buses along roads with one house per mile? Well, maybe you should ask the city to upzone. Oh, they say they can't upzone because there's no infrastructure for electricity or sewer? Well, maybe you should ask the Public Utility District. And then they would need schools? That's a problem for the Local School District. But the school district can't afford to run school buses everywhere? Well, that's nothing to do with the Transit Authority! Perhaps you should burn it all down, which will be easy because the levy to fund the local Fire Department failed and they can't afford trucks.
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.