The point is that stiffness is provided by reducing the degrees of freedom that would cause flopping to those that would require you to "stretch, shrink or tear" the piece of pizza. As a result, in cases where your stresses are negligible compared to the yield strength of your material this approximation accurately predicts the behavior without resorting to FEM or in depth analysis.

While I agree that there are more complicated theories that are correct for more diverse circumstances, I think it's tremendously valuable to find the simplest models that describe the easiest situations if only for the purposes of developing intuition. I must admit that this is very much a physicist's perspective, though.