Prices are decided at the margins. Having PE and REITs at every single table, even if their actual ownership is small as a %, makes huge differences.
They are responding to the market. If they overbuy then they will lose money and have to sell at a loss, at which point you could snap up some good deals.
The funds themselves say in their financials that they view housing as profitable because of the various restrictions on supply in every desirable city. They explicitly say that if those restrictions were lifted they would not be able to make money in that business and they would exit.