It could be substantial just by increasing the number of occupied dwelling units - if institutional investors are willing to let housing sit empty for 6 months or a year (because it doesn't hit them where it hurts) and the average vacancy rate in a city is hovering around 10%, dropping that down to 5% will really shake up prices.
If mega institutions are financial extractors who will take all money available, and the units can be rented to make more money, why aren’t they doing it?
Why would the institution turn away free money but individuals owning the houses wouldn’t?