I’m all in with your suggestion and want to point to housing cooperatives which are nonprofit by default and make the members co-owners of the complete stock of housing the cooperative is owning and managing.
Over here in Hamburg cooperatives handle about 20% of all housing dampening prices in general as they rent noticeably cheaper than owners who want to turn a profit (in Germany rents are bound to certain maximum levels defined by the market in the city).
Vienna has even more housing in the hands of cooperatives which definitely helps with housing and prices.
Oh you sweet summer child… It’s not an attack when the Führer commands it.