It wasn't PE directly buying houses that falsely inflated housing costs, but how they laundered their collusion through software to manipulate the availability of housing across the nation and manufacture demand for property, with global effects. They just set themselves up to massively profit.
PE-backed rental price fixing software allowed a handful of property management companies to control up to 80% of all multi-family dwellings in the nation. They were able to make their repayment obligations by setting non-negotiable rents despite massively warehousing apartments (some areas were maintaining as low as 75% occupancy rates, far below the historical 95% failure threshold).
If you can't increase the population, just artificially shrink the housing supply. And when rental prices are determined by median house prices, and house prices are based on the affordability of rentals... the feedback loop should be apparent.