Isn't the claim of (super)conformal invariance justified by the fact that up to the trace anomaly the trace of the stress energy tensor is zero? This is exactly what you would expect for a CFT and this is exactly what happens at the origin of the moduli space of the $\mathcal{N}=4$ SYM theory.

P.S. In general, if specific conditions are satisfied, for a scale invariant theory one can make $T_{\mu \nu}$ such that $T_{\mu}^{\mu} =0$ which implies the conformal invariance. To understand what happens with the trace anomaly is a long story that need to be carefully studied.