One approach is to replace the free abelian group functor with the free commutative monoid functor, also known as the infinite symmetric product.
Let $X$ be a pointed CW complex, and let $\mbox{SP}^\infty(X)$ be the infinite symmetric product. By Dold-Thom theorem, if $X$ is connected then the homotopy groups of $\mbox{SP}^\infty(X)$ are the homology groups of $X$.
The advantage of $\mbox{SP}^\infty$ over the free abelian functor is that it has a natural filtration, and a cell structure that can be analyzed. Given Dold-Thom theorem, the Huriewicz theorem is equivalent to saying that if $X$ is $k-1$-connected (for simplicity let's assume $k>1$), then the map $X\to \mbox{SP}^\infty(X)$ is $k+1$ connected. This can be proved by analysing the cell structure on $\mbox{SP}^\infty(X)$. You can find this in chapter 6 of the book of Aguilar, Gitler and Prieto.