FizzBuzz is an interview question you just expect.
RosettaCode has every FizzBuzz solution out there. Tons of people know how to FizzBuzz.
This is just "learning enough to pass the test" - a canned answer like the one you know you have for "How do you deal with multiple simultaneous high priority projects?" or "What's your biggest weakness?"
You may be filtering out lazy lazy candidates with FizzBuzz. But just because they can do it doesn't mean they can ship code.
I don't care what your biggest weakness is. You could have severe OCD or addiction. You could battle depression. You could be a chronic procrastinator. None of those matter. What does matter is if you give a solid straight honest answer, rather than some canned crap you read on a web site somewhere.
Same goes for how you manage multiple high-priority things. Every job I've worked on, every team, it always happens that there are multiple high priority things. Do you try to do them all and then crash and burn? Do you do them half-assed? Do you fight to get priorities aligned? Do you call the sponsors together and share information about the deadlines? Do you under promise and over-deliver? Do you ask for help from your team?
The way you answer is often more important than what you answer. Unless it's "Fuck off."
:)
If you don't know how to reverse a string, or split an array of ints into two with even and odd ints, or write a while loop that terminates on a specific word - I have zero belief that you can fill the position I have available.