There are many counter examples of companies paying equal regardless of the location and:
1. Not everyone lives where it cheap. There are many factors when choosing where to live: Friends, family, emotional attachment to a place, ...
2. That's linked to 1, if they choose to move somewhere expensive that's on them, not imposed by the company.
3. Not true, someone unhappy isn't going to stick around just because they can make more money.
...
At the end of the day the main goal of paying less where the cost of living is cheaper is to increase the company profit. It's 100% reasonable but masking the reason behind some "fairness" bullet points is BS to me.