If housing is an asset and its amount is constrained, then its value goes up. If I'm wealthy, I can use the leverage I have on existing properties to buy more properties, to earn a multiplier on whatever the market does. Have 10x properties? You get 10x the returns when the market rises. (And 10x the leverage to buy more property.) Not to mention you can charge more rent.
Once you reach a certain point, it might not be in your best interest for the housing supply to increase, as your existing investments won't be as valuable.
> The only alternative that doesn't have strictly worse incentives are if the government builds and assigns all housing.
It doesn't have to be all or nothing -- we can change the incentives in various ways. For example, tax on land value not property value. (Encouraging housing to be built rather than parking lots). Allow tenants the right of first offer on property they are renting, so an apartment complex can become a coop rather than owned by a private entity. Increase taxes on non-owner occupied properties, to encourage folks not to just buy and hold vacant property. Improve protections for tenants -- increase penalties for landlords who do not maintain their properties. There's probably a lot more ways we could incentivize increasing the number of people housed rather than maximizing the revenue for the landlord.
Build distributed public housing, in smaller, more manageable buildings. Don't require that public housing revenue covers all maintenance for the property -- if we agree that we should build some, why would we let it fall into disrepair? We should adequately fund its care.
In addition to tenant coops, support cohousing.