In this case, stopping or slowing this is actually pretty simple: Change the tax code which is currently overly favorable to real estate investors.
Specifically: No accelerated depreciation, stretch out the depreciation schedule to 40 years from 27.5, stop allowing 1031 exchanges, eliminate real estate step up tax basis on death, etc.
So the missing link is somehow proving the causality between "rents are high" and "3 companies own 19,000 rentals". That may seem obvious but could be very difficult to prove, especially as it may not be correct. Walmart owns 25% of the grocery business in the USA - does anyone really think grocery prices would go down if they divested? (It may still be better for society for them to do so, lowest price is not the only measurement, of course.)
First, most of these companies will hold large liability insurance protection.
But also, if someone sues and can prove negligence they can go further up the chain to the individual holding companies making the LLC shell game irrelevant.
The "best" might be to adjust property tax on a given description of dwelling unit such that it's pretty annoyingly high, but each family effectively gets one for free. Then corporations and multi-unit owners would pay that, but individuals wouldn't.
Prop 13 in CA would be much less of an issue if it excluded commercial real estate and only applied to "homesteads" owned and held by the resident owners.
I say this as someone who owned rental properties in an area that jacked up property taxes. Me and all the other landlords simply raised rent to compensate.
But if your property tax was 200%, and a home-owner's was 1%, things would change.
Now the argument about what the "minimal available housing" should be would come into play, because people who could not buy no matter what would now be homeless ...
I think we should remove carrots (the tax advantages at the federal level) instead of trying to penalize via harsher taxes.