This is utterly false. If a group $G$ acts nicely on a simply-connected simplicial complex $X$, then $G$ can be expressed as a complex of groups on the quotient space $X/G$. What you want is equivalent to having all torsion elements of $G$ fix points in $X$. It is trivial to come up with group actions on simply-connected simplicial complexes in which torsion elements act freely (though it may be worth remarking that this is impossible if the complex is either contractible or nonpositively curved).
EDIT : In response to your further question, here are some references. For the fact that a finite group cannot act freely on a CAT(0) space, it is an easy center of mass argument and can be found in Bridson-Haefliger. For the fact that a finite group $G$ cannot act freely on a contractible finite-dimensional space $X$, if it did then $X/G$ would be a finite-dimensional $K(G,1)$, which is impossible since $G$ has infinitely many nonzero homology groups (see, for instance, Brown's book on group cohomology).
It is maybe worth noting that a tree is both CAT(0) and contractible, so both of these facts generalize the observation of Serre you made in your question.