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.
That might be true, but there are lots of companies that own more than 1000 homes. 1-3 should be the limit.