Sure but now you're not just talking about UBI, are you?
You're now talking about UBI, plus rent controls, plus state-built housing.
All to make UBI actually do anything at all other than enrich existing landlords.
Why don't we just skip the UBI and the rent controls and instead just have the state build housing?
Because UBI is largely orthogonal to those things. It's a way of taking productivity gains and ensuring that the entire population benefits from them.