This proof is much simpler than the one usually presented in Galois theory classes, and in some sense it is more general - not only we show that the quintic is not soluble in radicals; in fact, the same proof also shows that the quintic is not soluble using any collection of reasonable univalent functions: $\exp$, $\sin$, $\zeta$, and even $\log$.
Yet one thing the classical proof does and we don't: Classical Galois theory can show, and we can't, that a specific equation, say $x^5-x+1=0$, cannot be solved using the basic operations and roots.