My understanding is that excise taxation is generally agreed upon as the preferred method for governments to manipulate the market, both because it incurs fewer deadweight losses and because it tends to actually work better.
So, what if we get rid of all these complicated rent control and rent freezes and affordable housing schemes, and instead just implement a rent tax, to be paid by the landlord, and make it progressive? I don't know exactly how it should scale; you wouldn't want it to be just by rent because that would have a regressive impact on families who have kids, because they need more space and more space naturally costs more. Maybe price per square foot?
At least in my city this would probably also reduce real estate prices in general, because a huge source of demand for houses is actually real estate speculators who buy up houses and then put them on the market as rental units. I gather, based on one conversation with an acquaintance who had been a realtor but was looking to pivot into this line of business, is that a lot of what's fueling that is, in effect, not-exactly-loopholes in US and local housing, lending and tax laws - many of which are ostensibly aimed at making housing more affordable - that allow people with sufficient resources to financially engineer together a speculative source of income while externalizing all the risk onto everyone but themselves.