No, you need to massively jack up taxes to increase supply, and it will work. But you don't jack up taxes on homes for rent, you jack them up on the homes that

No, you need to massively jack up taxes to increase supply, and it will work. But you don't jack up taxes on homes for rent, you jack them up on the homes that aren't for rent: houses which are just sitting unoccupied. The taxes will encourage the owners to either rent them out or simply sell them off to someone who will (or who will at least live there). That increases supply.