Rental is not necessarily the problem compared to leaving houses empty because they're worth more as investment than putting in the work to rent them out. And the number of Vacation rentals in my area is a bigger issue than rentals or empty houses. Why house community members for $2000 a month when you can charge that per week to tourists part of the year and make more money, right?
It's not just the home-buying market where prices are high. It's also the home-rental market.
> so that no one has to be homeless anymore.
People that can't buy may no longer be able to rent either because of the supply you constrained. Homelessness would probably go up.