Fizzbuzz wasn't designed to test people's prowess in coding. It was designed as "This is one of the simplest possible programs one can write. Can this candidate do it?" While many people may go through their whole career not dealing with dynamic programming or most data structures, almost all programming is fairly similar to FizzBuzz.
As an analogy, it's like giving someone who claims to be a circuit designer a basic circuit where he's failing at Ohm's Laws or Kirchoff's Theorems. Or an electromagnetics person who cannot integrate a basic polynomial.
If you have test anxiety that's one thing, but if your reason is "I need time to think this through", it's a huge red flag. Kind of like the person saying he needs to consult a table of integrals to integrate x^2.
I'm against intense Facebook/Google style whiteboarding. Fizzbuzz is at the extreme other end of intensity.