Unrealistic scenario is unrealistic. 5000 people didn't move from Saratoga Springs to NYC. And 5000 people on UBI aren't going to from NYC to Saratoga Springs, it already has expensive rent. No, if you're on a marginal UBI you aren't going to move to attractive places, you're going to move too someplace where you can stretch your money -- aka someplace cheap and someplace where you have support, AKA family.
People moved from tens of thousands of different American towns to NYC, and lots of people in NYC would love to move back to those tens of thousands of different towns. And if you move back to where you came from, it means that you're not going to stretch the capacity of where you're moving to.
5 people moving from NYC to 10,000 different towns is far more realistic than your scenario. And 50,000 moving from NYC is likely going to affect rents there more than 5 people moving elsewhere.