It's not that simple - the problem is that those institutions are market makers. They are a tiny portion of the market, but a huge driving force in setting and manipulating prices, because their properties get leveraged, instrumentalized, and securitized, with derivative products, speculation, and all sorts of incentives that you don't normally want operating in the arena of housing.
The things that they do have massively outsized downstream impact contrasted against their relatively tiny overall participation in the market, and they can afford to behave in ways that manipulate the behavior of the majority.
If you can decouple them from the housing markets, you also decouple the interests of the donor class, and you allow for policy that doesn't maximize the cost of real estate over the interests of the majority of the population.
The things that they do have massively outsized downstream impact contrasted against their relatively tiny overall participation in the market, and they can afford to behave in ways that manipulate the behavior of the majority.
If you can decouple them from the housing markets, you also decouple the interests of the donor class, and you allow for policy that doesn't maximize the cost of real estate over the interests of the majority of the population.