This is basically an extremely classist, racist and uninformed version of NIMBY that says "Poor people: They are allowed to exist, but Not In My Back Yard. Go be poor someplace else!"
It's a complicated issue and I've already written a couple of long comments and I have other things I need to be doing. But telling people we will cut you a small check so you can move someplace cheaper amounts to saying poor people cannot live in desirable areas with lots of amenities and have no right whatsoever to desire to do so.
I also have zero reason to believe it actually solves the housing affordability issue. I was willing to move to almost anywhere in the western US to get off the street and back into housing. I left California and moved to a small town in Washington. But it was extremely challenging trying to find a place to make my life work with affordable rent and no car.
I no longer drive. Even if I did, buying a car is typically the second highest household expense for most American households.
We just don't have a lot of options for a full and vibrant life without a car in the US. Living without a car requires a certain level of urban fabric so you can do your shopping etc on foot and via public transit. The lack of walkable places in the US is a primary part of the affordability issue.
It will not get solved by injecting funds into the system for poor people to use to cover rent. In fact, doing so likely makes the issue more intractable.
Real change is always hard. People inherently do not like change. UBI is a "status quo is god" answer when the current status quo has terrible problems that will not be fixed by simply giving poor people a few more bucks to live on.