That extra 200 will also allow some people to move to a rural area, decreasing demand, which means you won't find a renter.

Yes if you simply assert an upside down reality, this is a good solution.

However, people actually move toward higher COL areas as their income permits them to.

If more income meant people moved away from high COL areas, cities wouldn't exist. We'd have a flat distribution of people across approximately all land with ultra-low COL and ultra-low productivity everywhere.

This - since you can live in a rural area with UBI - and you get more time in the day to manage your accommodations, the move to urban housing is not so critical.