Amazingly enough, pension funds are allowed to park money with them. For whatever reason, unlike endowments, which are heavily restricted in what they're allowed to invest in, pension funds can invest in nearly anything they want to.
But they're still called private equity, which has nothing to do with who is allowed to give them money to invest.
So, it's not really "you will own nothing and be happy about it", but rather "you will own 0.1% of 1000 houses rather than 100% of 1 house, and be happy about it."
If only that were actually literally true. If I could buy shares in a housing cooperative that allowed me the right to live in any house owned by the cooperative, and when I owned shares equivalent to the market value of a house I could trade those in for full ownership of one house, that would actually be pretty interesting.
Honest question from outside of the western world: is it really private, as in, the end beneficiaries are "evil rich" people with net worth over $10m? Or are those investment funds that hold the pension plans and life savings of the western middle class?