Because the point of FizzBuzz is not to get a working FizzBuzz class that can be integrated into an application, it's to see if the candidate can actually produce a working version themselves.
Imagine you're hiring someone to do your taxes. Sure, you'd expect them to use a calculator to do all the hard addition/subtraction, or more likely spreadsheets. But if you asked them to add 5 and 3, and they said "I would use a calculator, this is a stupid question, I can't just tell you off the top of my head" then you're probably not going to hire them.
It's a super basic test of 'Can this person code themselves out a paper bag."
which can be determined entirely without the aforementioned methodology. if you read the guys article, you'd see that he'd already completed the basic tests demonstrating basic competency.
being able to solve problems is more important than how you solve problems. (generally speaking, yes i know there are exceptions)
And that's why Amazon's phone-interview coding exercise is a rather complicated coding question delivered under proctoring, rather than simply fizzbuzz itself: to actually filter those people out early and avoid the cost of flying them to Seattle, rather than just "going through the motions" of doing so.