clarification on why BG has no action

Question

I got confused by the page (Are these two definitions of $EG$ equivalent?) about the following.

Doesn't this also define an action on BG?

Answer 1

A $G$-simplicial set $X$ is cofibrant if, and only if, each $X_n$ has a free $G$-action. The only sensible $G$-action on $BG$ is the trivial $G$-action. But the trivial action is not free: every simplex is a fixed point! Thus, $BG$ is not cofibrant as a $G$-simplicial set.

Your confusion might arise from trying to see the object $BG$, whose natural category is $\mathsf{sSet}$, inside of $G-\mathsf{sSet}$. A different question is whether $BG$ is cofibrant in $\mathsf{sSet}$, and this is indeed the case: all simplicial sets are cofibrant.