Indulge me one step further. I assume you want these to be non-profit because they would be offered for rent for less than full market value, at what you consider a non-exploitative price. This would place many rentals on the market at below-market rates. In other words, many people would be willing to pay the asking price. How would you decide who gets these rentals?

The whole point of a free market is to allow competing offers to set price. If you set the rental rate on a beachfront condo in Santa Monica that currently rents for $5,000 per month at $1,000 because that reflects the cost for a non-profit to maintain the abode, you would have thousands of applicants willing to pay that price. Who gets it? Does some committee decide who "deserves" it? What is their criteria? Is it income based? Do maids and landscapers qualify for beachfront condos? Is it rented to the one who bribes the selection committee (which would happen)? The one who writes the sappiest letter? The selection process would quickly become corrupted.

In my opinion, fair market rent is the fairest way to decide who gets what. That isn't some landlord being greedy or exploitative. It's the activity of potential renters putting in bids to set the price. This isn't exploitation. If some landlord wants to get greedy and raise rents above market, his unit will sit empty. He will pay the price for his own greed and stupidity. That's how markets correct for that kind of thing.