Nonetheless, I am amazed that every textbook I have seen uses the "optimize a quadratic form on the unit ball" argument rather than this algebraic once. Lots of students don't remember multivariable calculus well, and existence of maxima of continuous functions on multidimensional bounded domains is complicated. Plus, I find a lot of students have trouble with an inductive process like getting one eigenvector and splitting off an orthogonal complement.
This argument is just shuffling algebra around, combined with the fact that a sum of squares is nonzero. It seems clearly easier to me.