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
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.