You might want to read into data like this (1) to get a better view of how much housing stock those companies own as a proportion of the total market (it's very little). The data is kind of hard to get, because many small landlords have incorporated as LLCs/LLPs and they make up the bulk of "corporate" ownership.

But it's pretty clear that the big time owners like institutional investors/REITs/etc own less than 2% of all units.

(1) https://sgp.fas.org/crs/misc/R47332.pdf