The denialism of market forces is one of the worst aspects of progressives. Like it or not, the only way housing is going to become cheap is if there so much housing available that landlords actually have to compete for tenants.
You can build public housing, and have a 20 year waiting list for it. You can have rent control, but that only benefits long-time residents at the expense of newcomers. But the only thing that will make cheap housing widely available is to make it easy to build, so developers can saturate the market with housing.