We haven't built enough housing for our current population. Private equity is buying housing because it's a valuable resource. If we didn't have a lack of housing, they would move to something else.
Granted yes we should get rid of private equity because it doesn't encourage good practices (dumping debt, doing the cheapest thing to make a buck, etc), but would rent be lower if housing was owned by regular large corporations or landlords? Greystar is the largest apartment owner and it's not private equity but had to settle a case for price fixing, as well as RealPage which helped landlords coordinate their rent prices. There's asymmetries of information and need between landlords and renters.
The need is obvious because a large landlord can hold a home empty longer than I am willing to sleep in my car, much less a person with kids or someone needing assistance like an older or disabled person.
But the information difference is important too. How often do you look for a place to live? Once every few years? Landlords do it way more often. They know better what's on the market, what the rates are etc.
If you limit the number of units anyone or company can own, that would solve a lot of that regardless of if the company is private equity or not.
But if we build all the housing we need for today, then we've drastically overbuilt for the future if the projections of a coming population decline are accurate. We could retrofit some office space, but that's more complicated than it appears largely based on plumbing and requirements for windows and such.
But that could be done. If we do decide to build more housing that runs into NIMBYism and that it will detract from a lot of people's equity. We could build in a multipurpose fashion if we address zoning which doesn't seem like a big blocker. Massachusetts and California are trying to force cities to allow multifamily buildings near public transit, but some cities are resisting because their citizens hate the idea. And the option more common in Republican states of building outwards has major issues of people spending time stuck in traffic and physical space limits in a lot of places and lack of public transportation.