Adjusting by CoL is only good for the employer, if a worker is worth X working remote in SF they're worth the same amount in the middle of nowhere.
But the competition tends to be fiercer in a higher COL area, so you are still not guaranteed even a median pay job in that area if you cannot get it. Many do not. The "higher COL area = higher pay" equation seems magical except it hides that it also includes higher competition.
Of course the competitiveness matters. It informs the level of optionality and leverage they have to negotiate. I would say that it's the only thing that matters. What about the CoL adjustment is weird besides it being reflective of the somewhat ugly and distasteful truth that to the company, you are a human resource and fungible cog? That's what they're paying for.
This employee will have to think about what happens if they do not come to a favorable agreement with their employer. Their BATNA completely depends on 1) the demand of other firms in the area (or remotely) and 2) their relative ability to compete. If they can perform well enough to move to a competing employer in the same locale that pays better, they will.