I think this conclusion is wrong. The fact that FizzBuzz exists actually means that programming interviews are awesome: with most other professions, you would end up hiring a candidate equally as incompetent as the programmer that failed FizzBuzz.
No. You do not have to ask doctors or lawyers or taxi drivers fizzbuzz levels of questions. If applicants for doctors jobs were as incompetant as programming interviews, you'd have to ask applicants to location the ear on the human body. Doctor applicants aren't that incompetant, programmer applicants are.
I've had plenty of taxi drivers who could barely drive, let alone find any given address in their city. And as for doctors, see my answer to potatolicious above - I've seen some terrifyingly incompetent ones who really should have been screened out with a fizzbuzz level question.