I feel like there's an implied "and keep their comp". There are a decent number of remote jobs out there right now. They pay well, but certainly not SF TC.
--
Say you move to a lower cost of living area, perhaps 6% cheaper, and your employer adjusts your wages down by the same amount as the change in COL.
Now all your expenses are 6% lower, but your money left after expenses is also 6% lower. That means that despite being no worse off in terms of quality of life, your savings went down 6%.
Strictly speaking, if you maintain the same relative ratio of costs to savings, it’s always rational to take the highest income, even if it’s in a higher COL area. This is because costs can be measured as a percentage of income, but savings ought to be measured as an absolute value.
--
The "fix" would be to somehow work out what workers spend, and only adjust that part of their salary by COL. That's invasive and gives an unfair COL-adjusted comp advantage to the highest earners who can save a bigger portion of their income.
I don't have an ideal solution. I guess there could be a standardized single remote COL adjustment (perhaps adjusted by country or something to account different legal frameworks and mandatory benefits), and the remote worker can live wherever without comp adjustments other than local tax.