I don't see how that really matters. If you keep making homes and investment buyers keep buying them then the price will keep going up and regular folk will still be priced out. If you make so many homes via this strategy that institutional buyers are no longer interested, I think it's because the homes have somehow become genuinely worthless, at least to the investors. I don't see how to do that without legislation that makes the investment unattractive (unprofitable) to them. Policies such as rent control, "you can't buy that house because you're not living in it", etc., are the proper solution.
Investors have finite capital, and it's easier to corner a tight, pinched market than a large market with lots of liquidity.
You don't see PE making huge investments in gas stations, because you can always easily just go to another gas station, and there's way too many of them to easily corner the market. We could pretty easily make housing similarly unattractive.
Why would the price keep going up if more and more houses are being added? Prices go up when there is a shortage. Everywhere that has been building adequate housing sees prices falling.
With greater supply, there comes a point where you can make better investments than buying up empty houses. And so they will stop buying them.