Where would this money come from, though? You only make money on the property when you sell it to someone else, so who is buying it? Not the next renter, because they are supposed to be buying into equity, not paying someone else's profit.
I think there's some cognitive dissonance in the proposal. The idea that you can live for free is just not possible in the modern, capitalist world. Rent is something that every entity has to pay - including owners (which is the imputed rent, or the opportunity cost of the capital placed into the building/land, or the interest payment).
If the gov't fronts the capital to build, but don't receive rent, then it's the same as tax payers footing the interest cost of that capital. So those who would be in this public-ownership housing is effectively receiving an interest free loan for the house, and their payment represents equity, and that's such a good deal that there'd be unlimited demand for this sort of housing.
You can't solve skyrocketing housing prices by simply increasing access to capital... that is just going to exacerbate the problem
Do the neighbors vote on their next neighbor? The city council? An impersonal bureaucracy? What factors does it consider? Are there priority quotas for people meeting certain criteria? What criteria? How many people? Who decides? Who decides if an individual meets the criteria? How do we keep them honest? Can their decisions be appealed, can normal people afford to appeal them?
I don’t think it’s impossible to come up with good answers here, but that’s the work. The mechanism design here is extremely important, and it’s not “simple.”
A simple solution congruent with these principles could be:
Selling the house should be restricted to yield the amount of money you put in. This excludes money spend on repairs, etc., and is to be adjusted for inflation.
You can keep living in the house as long as you are alive. Once you are dead, you lose the house and the inheritance will be the same money you would get from selling the house/apartment under the regulated terms.
The next renter of the regulated public housing apartment would just pay rent again until they have enough in their portfolio and then stop to have to pay rent.