Imagine if we did the same thing with, say, water: privatized it and then "democratized" ownership of it.
It's easy to get retail investors to buy things at inflated, speculative prices, so the price of water would go up. People would say, "Why are so many people dying due to lack of access to water?" And we would blame these startups, laws would be passed, and we would fix it.
Unfortunately the necessity of housing is not as direct and clear, and we have a long history of denying housing to people who are unlucky, sick, or the children of such people.