Then tax land. Make landlords sweat if they fail to rent out units.

The solution to housing affordability is all about negotiation power. Currently, tenants in many cities do not have negotiating power because an overall shortage of housing and thus no credible threat of vacancy (and with no land tax, vacancy isn't enough of a financial loss). Build more housing by making it legal and easy to build housing, tax land, and tenants will have far more negotiating power.

A good example is my city, Montreal. Thanks to being one of the most YIMBY cities in North America, we have far more missing middle housing than any other city of this size on the continent. The result? More power to the tenants, less to the landlords. And the result of this power? Not only is my rent waaaaay cheaper for what I get compared to my sister in Boston (we pay about the same, but she has 3 roommates and lives farther out, while I have 1-br and live right next to downtown in an insanely walkable neighborhood), but I also negotiated down my rent when moving in. The reason I could negotiate down my rent was because the landlord had a credible threat of vacancy, so they'd rather lower rent by a hundred bucks than risk vacancy.