Well, I am not sure if it can count as a proper proof but you can see that the β-function vanishes at 1-loop level by knowing how the various fields contribute to it in an \(SU(N) \) theory. You do this by evaluating the (Dynkin) indices of the various representations (Weyl fermions, complex scalars and gauge fields) inside \(b_0\)and see what happens. You can find the Dynkin indices in Terning's "Modern Supersymmetry" Appendix B.
Furthermore, another simple argument on the vanishing of the β-function in N=4 SYM is given in the same book in section 14.4, page 241.
I hope this helps!