Outside of a few metro areas where institutional ownership is very high, I don't think this would change anything. As long as houses remain an attractive investment, non-institutional smaller investors will happily buy the properties for a few thousand dollars less than the institutions would.
Anyone familiar with basic economics is pulling their hair out reading this, because there's one extremely obvious way to lower the price of building new housing: Reducing or eliminating tariffs on construction equipment and materials and ensuring a robust supply of low-cost labor.