I don't think so. In theory it sounds great, but it's a little bit of the power fantasy "what if I was the landlord?"
The lack of a profit motive helps but you still get taxed on a growing property value, and you probably have to pay at least some of the people...there's always going to be repairs needed and plumbing to be fixed and a new roof every decade or two. And since those prices are going up, rent will keep going up by necessity.
So it's a lot of work and fundraising to provide a good deal for not actually that many people. If a community gets priced out of a neighborhood except for the 4 people in your multiplex, you haven't really combatted gentrification much.