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.