Of course buildings cost money, no one is suggesting that the tenants of these public housing buildings don't pay rent. Just that the rent is set at a rate that simply recoup costs instead of making the landlord (which in this case is the state) rich in the deal. And if states don't have the funds to invest in the property, I'm pretty sure the state governments would qualify for some pretty solid mortgages. And the costs of those mortgages can be added onto the rent - same as a landlord would do, only in this case, that would be the end of the rent padding.

I can't help but feel like you're deliberately misinterpreting me at this point. That, or you're just incapable of fathoming how human beings could possibly interact without one profiting off the other. The renters still pay rent, the mortgages on the property still get paid, the only difference is that the profit that would have gone to the landlord stays in the tenants pockets. That's it - that's literally the only difference. No free houses, no huge tax bills, just the removal of profit, and at-cost rents for folks who need them.