The rent prices are high (low supply), so whoever has the most money gets it. If you only have 1000 from UBI, but the guy next to you has 1000 plus the 1000 from UBI, guess what the new price is?

Build another house, and then problem solved. Why start with the assumption that supply needs to be low forever? That is actually a very new problem in housing due to regulatory capture, a long history of terrible zoning laws, and a tragedy of the commons that there are always many people ready to oppose a new house but no resident of the unbuilt house able to counter that force and argue for it.

With better policy we really could end the current housing shortage in our biggest & richest cities.

But in America there are more empty houses than there are homeless people. Supply is not low. Maybe rent would go up in some markets (eg Bay Area), but people can move somewhere else. That’s the point of having a market system.

Yes UBI decreases market inefficiency, but the effect would be smallest on inelastic markets like housing. You’d see the biggest impact in the markets for the life-improving items/services that people with less money can’t afford right now, everything from fresh/healthy foods to nice shoes. The prices of those things would go up, but it would reflect an increase in demand.