Demand isn't high because so many more people prefer to rent - demand is high because it's the only financially viable option. Why is it the only financially viable option? Because landlords (both corporate and personal) buy up all the property they can and rent it out. Because so many houses are getting bought as rentals, the supply of houses that can actually be bought is low.
Seriously, have you spoken to anyone who has tried to buy a house in the last few years? Every single one I know had a myriad of stories like "I put down an offer, but some investment company offered $20k over asking, cash in hand"
And because housing prices are so high because of the above behavior, more and more people are forced to rent, who would have 100% been able to buy a house not that long ago. And so rises demand.
If what you were saying was true (that rent prices are high purely because people love renting, and no one wants to be a homeowner), then why are we seeing sky high home prices at the same time? You're quick to pull out a half baked supply and demand theory, but you're very quick to ignore the other side of that equation.
Also, more fundamentally the whole "supply and demand explains all commerce" thing has been thoroughly untrue for ages. Maybe in a world without giant multinational conglomerates, political corruption, and price fixing. But in the real world, things are wildly more complicated
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.