In Seiberg and Witten's seminal paper, a key role is played by the monodromy of $\tau$ around infinity. This monodromy can be computed in the weakly coupled regime, and it is given by the one-loop beta function. It is important, as stressed by the authors, that higher-loop corrections vanish (and the full answer is only modified non-perturbatively, by instantons).
Now, the fact that the higher-order loop contributions to the beta function vanish seems to be a consequence of supersymmetry. Indeed, Seiberg's holomorphy argument shows that physics must be holomorphic in $\tau\sim\theta+i/e$. But perturbation theory can only see $e$, so it cannot generate holomorphic corrections. Thus, these corrections must vanish.
We are often taught that having a beta function that vanishes beyond one-loop is a rather trivial statement. Indeed, 't Hooft showed that these higher orders are scheme-dependent, and can be made to vanish by a convenient choice of renormalization scheme. Thus, the fact that the beta function in $\mathcal N=2$ is one-loop exact does not seem like a very deep statement: one can always choose this to be true, for any QFT, regardless of supersymmetry.
So, on the one hand, Seiberg-Witten insist on the importance of having only a one-loop monodromy; and, on the other hand, we have 't Hooft argument that this is a trivial fact. Needless to say, both claims are definitely true, and I am just missing a piece of the puzzle. What is special about the beta function of $\mathcal N=2$? How is it different from that of regular QFTs?
