For a simplicial group $G$, there is a universal bundle $WG \to \overline{W}G$ in the category of simplicial sets, detailed in for example May's book (djvu).

Now $WG$ has a simple enough description in terms of $G$ that I would expect one could construct a contracting homotopy directly. Has this been done in the literature?

The proof in May's book (and in the original sources) that $WG$ is contractible goes via showing that $WG$ is 'of type (W)', and that such simplicial sets are Kan, simply-connected and have trivial homology.