Fizz-Buzz (or equivalent tasks) are just a quick way to tell whether the minimal basics are met. I don't ask Fizz Buzz but give problems that are only slightly more complex.

I'd consider it to be a "handshake". If the basics don't work you don't need to waste both sides time.

> People who can't code fizzbuzz will usually fall flat on those questions as well

yes, but it is harder to tell whether they were completely incapable (what FizzBuzz will tell you) or whether they just didn't touch certain topics in their career. I'll gladly hire a capable engineer who was active in a different domain who I am certain will be able to grasp the necessary knowledge quickly.