If the answer is "no, of course not", then why is it okay for them (or any company) to pay you less, if you move to a lower COL area? It absolutely isn't fair, and companies want to eat their cake and have it too.
If you actually transfer to an SF team then yes, most large, multi office companies would adjust your pay to the SF bracket. If course that requires that an SF team has headcount and wants you. You can't just move because you feel like it.
I'd be surprised if this isn't the norm in companies with a distributed workforce.
And you win the Nobel prize for stating the obvious.
Tech salaries today in hotspots like Silicon Valley are probably on the order of twice what they were in the early noughties. Why? Because everyone’s so much more productive these days? Or because that’s what the market dictates?
Frankly, the calculus of your pay probably depends a lot less on your knowledge or output than you seem to think. Geography and luck play as large of a role.
I tend to think I should be paid X and should I choose to live in a more expensive region, that's the price I pay for choosing to reside in that region. If I decide saving million dollars is more important to me, then I move to Cleveland, maybe don't have the same scene as SF, but am able to save more, my choice.
Why should people in SV earn 120k+ per year over someone out in the rust belt doing the same work at 50-60k a year?