I see, that the trend to increased rent prices balancing out UBI would apply to in demand regions, but not at all in areas that would otherwise go unused. Most countries do have an extreme oversupply of living space, even before a population decline.
The places with highly desirable real estate and high prices (e.g. SF and Manhattan) are already unaffordable to people making less than $80,000. So UBI would not increase the ability for residents to pay rent in those areas. Rather, it might actually decrease their net income, so by GP's argument it might lower rents in those areas. But in small towns (like City You've Never Heard Of, WV) with a low median income, UBI would increase net income. So it might increase rents there except those small towns are already emptying out as people leave to seek high-paying jobs in big cities, as you said.
So I think UBI would lead to rental/COL costs homogenizing somewhat rather than simply going up. It seems like this could solve several problems at once.
The biggest problem I can see is housing, if housing supply is constrained then I suspect any UBI will just end up in the pockets of landlords. Perhaps universal housing would be a better idea.
People bitch about farmer subsidies but without them, farming would be a much more volatile business and farmers would be much more unlikely to be in the business. Thus cutting supply and making food much more expensive.
I wouldn't classify it as being naturally oversupplied. It's a result of the way governments manage the food production.
But there are some items that are in oversupply that would always be affordable so giving someone money to buy it will probably help. I can only think of virtual goods as goods that are always in oversupply. A place to live is not one of those items.