Which doesn't really matter because they don't need to be "protected" in any capacity. Removing their ability to purchase those homes, and letting it get sorted between the people who are "outlawing building homes" and those who want to move there makes more sense.
In this case, by restricting who can buy, you're reducing the demand for housing. Less demand means lower prices. Lower prices mean fewer homes will be built, meaning less supply.
A better scheme would be to lower the costs of home construction by reducing regulations and taxes on them.
The issue with housing affordability is that the market is often prevented from responding to high demand and low supply by regulation. SF is very restrictive with permitting housing construction, despite the incredibly high rents. It's not that the market forces are failing to incentivize housing construction, it's that developers are prevented from responding to market demand.
You're assuming a baseline of a functional market system.
If you have widespread price-fixing between landlords (say, via everyone using the same algorithm service that gives everyone the same recommendation) then rent control isn't introducing a price fix, it's just changing the level of price fixing.
If you can only rent for $X000, but the rent price is fixed to at least $X500, then for you, there might as well be no availability - the landlord is leaving the place vacant due to the price-fixing.
This is called a cartel. The trouble with cartels is enforcing it. Cartel members cheat all the time by selling for less than the cartel price in order to get a larger share of the market. The setup is unstable as the incentives are much like the Prisoner's Dilemma.
Markets are powerful tools--perhaps the most powerful tool we have to shape human interactions. But too many people believe in a sort of market gospel ("Supply Side Jesus"); that market forces are as inevitable and inflexible as the force of gravity. In reality, policy choices shape markets and careful market design can deliver politically-favorable outcomes.
Markets work for us, not the other way around. "You shall not crucify mankind upon a cross of gold!"