Governments have done the work of providing housing in the past, and still do in limited numbers. There is no reason why they can’t just push the number of projects up until there is no housing crisis anymore.
I’ve heard some numbers here and there, and it seems like there’s plenty of organizations providing non-market housing that rent at below half the usual prices. Apparently the YMCA is one of them.
If the governments aren’t willing to do it themselves, they can just make it easier for corporations that are willing to provide non-market housing to get the property rights and loans needed to actually get this done.