If PE comprises 44% of purchased homes, they're increasing demand, and therefore prices. So they have plenty to do with the current housing market.
Don't let idealism and perfection be the enemy of good: PE is currently distorting the housing market. The government should take steps to prevent PE ownership of housing, either through taxation or blanket bans, because civilisation imperils itself if most people can't afford housing.
I'm sure other measures should be taken to prevent housing from ballooning in cost between generations, too -- we probably want to stabilise it to something resembling inflation (but probably not necessarily locked to that measure?). That's for economists to figure out.
All I know is that housing has skyrocketed in price in the last few years to the point where many will never be able to buy, and more and more are priced out of even renting. Kicking PE out of the market may be a bandaid on a bullethole, but it's better than doing nothing.