I do agree with the second half of your post. I think, for the average lower-class working Joe, I'd like to see a mass increase of supply and then some kind of government voucher system to aid in affordability. This preserves the freedom to choose and move and doesn't lock the poor into Le Corbusier-style towers.
I said "possibly" because I believe there are still going to be a group of people that needs government-built housing, mostly the mentally ill/those who cannot live on their own and don't really have the agency to go through the home-buying process. I prefer this to nonprofits because government agencies can be held accountable easier than vague boards of directors. Plus, in my experience, non-profits are difficult to work with and are often poorly run.
In an ideal world, we'd increase the supply so much that prices relax naturally and the voucher system is not needed. In the urban planner world, we're experimenting in a few different cities with different ideas like converting vacant storefronts to housing. Office buildings seem like a great idea, but the problem there is that the infrastructure is simply not there for 200 new families and all their basic needs. It's bonkers expensive.