If $X$ and $Y$ are $G$-spaces for any group $G$, then the projection $\pi\colon X\times Y\to Y$ is a $G$-map, trivially: $\pi (gx,gy) = gy$. The map $\pi_2$ of your question is a very special case; note that $\mathcal M$ as I defined it is an operad as I defined operads, with $\Sigma_j$ acting on $\mathcal M(j)$.
Model categories are extremely important but entirely irrelevant here, and taking cofibrant approximations of operads tends to destroy their relevant individuality: different $E_{\infty}$ operads play seriously different roles in the applications, which are what people should care about.
Also, by my definition, $E_{\infty}$ operads are $\Sigma$-free rather than just locally contractible, so $Com$ is certainly not an example; $E_{\infty}$ spaces are far more general than $Com$ spaces, which of course are just commutative topological monoids.
This edit is to answer Lano's clarification of his question. Lemma 3.7, p. 24, op cit shows that a local equivalence over $\mathcal M$ is necessarily a local $\Sigma$-equivalence. The statement of Corollary 3.11 follows.