It's not a matter of should. It's just that, on average, someone in a cheaper city will accept less than someone in an expensive city because what is too little to cover rent in some places allows you to live like a king elsewhere.
It's just supply and demand. Give 50k in a poor country, and (assuming there are enough qualified people), you will have quite a lot of choice. Give 50k in SF and people will be laughing at you. Why give 200k in the first instance when there's enough supply with 50k?
There's nothing legal about that. If you claim that you live in SF to get SF salary but live in Alabama. You are committing tax fraud. You need to file state taxes based on your location. If you live in both places, tax forms allows you to set how long you have lived in both places. You will be earning the benefits of Alabama but paying no tax to them. That's why it's not legal, so if you wish to be legal you shouldn't lie about your location. Also your unemployment tax will be paid to SF and not to say Mobile, Alabama. Outside of taxes, different states have different rules on say over time, paternity leave, termination process, non-compete, discrimination laws that companies need to be compliant with.
You're not getting less, the world doesn't have a global currency with fixed price. It's all about earning power. If a company has an office in Alabama, and another office in San Francisco and another in Accra, Ghana. Employees won't expect to be paid the same, so why should they expect the same because they are remote?
It's just supply and demand. Give 50k in a poor country, and (assuming there are enough qualified people), you will have quite a lot of choice. Give 50k in SF and people will be laughing at you. Why give 200k in the first instance when there's enough supply with 50k?