Let's see if innovation, driven by greed, can change it.
This is similar to a co-op where the land is owned by the building as a whole.
You can't get away from the problem that if you price something under the market price, you'll get a lot of people who want to live there. How do you choose who lives there? Lottery? There's a huge incentive by the people deciding to self-deal.
Even then, it's still not efficient. For instance suppose the market price is $200k and you give it away to someone who really needs it for $100k, you just essentially gave them $100k. But they have to accept that benefit in form of housing. Perhaps they would prefer to live in a home worth $150k and get $50k in cash for groceries or school or whatever else?
And once they're there they won't want to give up the benefit. They're essentially getting something for nothing. Anyone who grew up in a city with rent control can tell you stories about people they know exploiting the system.
All these efforts to control price are like trying to control global warming with new thermometer designs. The price is a signal of larger economic issues. The only reasonable solution I've found is allowing people to build more.
For the issues being raised here (home ownership is inaccessible to many, renters don't have stability, security), aside from greater protections for renters, another class of existing solution is Community Land Trusts. These organizations retain ownership of the land, have a charter that's targeted at keeping housing affordable (or some related mission), lease out land on very long terms (like 99 years) with contracts that place limitations on resale of any improvements on the land. These restrictions can make housing permanently affordable, and removes housing on that land from being an asset for speculation. Residents get security from the very long terms of their loans, the governance of their CLT, and ownership of improvements on the land. The CLT can use revenue to create and maintain community amenities.
Sometimes good arrangements already exist, but they're not going to be rapidly scaled up across the country because no investor stands to get a windfall from rapid expansion.