It’s strange that you admit the interview technique doesn’t work well for people like you, yet you choose yourself to administer a similar technique and see “value” in it.
Why do you think you’re in the minority? Belief that whiteboard coding puzzle evaluations are broken is very widespread. Even many people who do not experience anxiety from the time pressure or “gotcha” nature of these interviews also see them as a scourge of the industry.
San Francisco is a large place with many companies. Better to find the ones that really fit your needs than to compromise yourself by trying to fit into an interview procedure that isn’t right for you.
Plus, since you mention you already have a job, it means you can take your time to filter out companies. No need to rush or apply pressure to yourself to hurry up and meet some interview standards... you can take a longer, careful search before relocating.