I don't think liberal zoning is the problem. I've seen my city go through massive changes in the last 30 years, top of the list being a radical increase, a liberalization, of zoning. They are allowing more and bigger developments. Multi-story and multi-unit developments are everywhere, but that isn't making housing any more affordable.
A piece of land with a liberal zoning is worth more. It is a target for redevelopment in order to max out the zoning. The net result in my local is the steady replacement of anything old (ie 10+ years) by newer ridiculously large replacements. Apartments and other attached units are now so expensive, and prices rising so steadily, that the logical thing to do is let them sit empty. The pittance one might claw back in rent pales in comparison to rising prices, and who needs pesky tenants anyway.
So... liberal zoning --> rising land values/prices --> redevelopment is better than renting --> fewer rental units = higher rent. It's true up and down the west coast. Now if a government told owners that no, they are not going to rip down that apartment block to replace it with luxury living, all-one-level, condos (ie for old/rich retirees) then perhaps prices might stabilize, sending more units into the rental market and so also stabilizing rents.