Oh, come on! Prop. 17.4, p. 69, of my ancient but still current book
``Simplicial objects in algebraic topology'' proves that the homology
groups of the Moore complex of a simplicial group $G$ are the homotopy
groups (defined simplicially) of the Kan complex $G$. That G is a Kan
complex is Thm 17.1. The homotopy groups of any Kan complex agree with
the homotopy groups of its geometric realization (op cit, Thm 16.6).
The cited book is still available from the University of Chicago Press
(or Amazon of course).