Before we even reach the question of how true that is, there isn't evidence that any firm holds a significant portion of the "houses on sale". A starting point here would be the fact that corporate investors buy houses and hold on to them, and thus definitionally don't hold any of the house on sale, but whatever, either way, just flesh the story out instead of handwaving it.

if we start with reasonable but definitely vague numbers that suggest 2M houses are for sale and institutional investors own 500k of the total stock, it suggests this is NOT true; it's unlikely they own all their houses in the same geo market and they're all for sale at the same time. This doesn't mesh with a business strategy (diversification) or the typical model (they rent houses; they don't flip them).

Doesn’t rent have a constant influence on house prices? Along with the data science based rent prices they demand, that implies a constant upward influence on rent and consequently house prices.

Also these institutions would be buying houses in high demand areas.