Vidit's comments are relevant to a paper of Graham Ellis

Homotopy classification the J.H.C. Whitehead way.
Exposition. Math. {6} (1988) 97--110.

The more general result given here is that if $C$ is a crossed complex and $X$ is a CW-complex then there is a bijection of homotopy classes

$$[X,BC] \cong [\Pi X_*, C]$$

where $BC$ is the classifying space of the crossed complex, and $\Pi X_*$ is the fundamental crossed complex of the skeletal filtration of $X$. A cubical version of the proof is given in the EMS Tract Vol 15 Nonabelian Algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, where a pdf is available. This book also contains our generalisation of the work in CHII on the relation between crossed complexes and chain complexes with a group of operators. He writes in essence that the former have better realisation properties, and the latter are better for calculation.

Note that the construction $BC$ generalises Eilenberg-Mac Lane spaces, including the local system case.

This has been generalised to the equivariant case,

Brown, R., Golasinski, M., Porter, T. and Tonks, A.
Spaces of maps into classifying spaces for equivariant
crossed complexes. II. The general topological group case.
$K$-Theory 23 (2001) 129--155. arxiv 9808111

There is a lot in Whitehead's CHII ! One example is his result on free crossed modules. I wrote up his proof ([30] in my publication list) and the referee wrote that: "The theorem is not new, the proof is not new, and the paper should be published." This theorem is sometimes stated in texts on algebraic topology but rarely proved. Note that Whitehead's "homotopy systems" are our "reduced free crossed complexes".

To go back to the original question, it must be said that the homotopy groups are but a pale shadow of the homotopy type, and indeed crossed complexes give only a "linear" perspective, though linear methods are often useful as an approximation.

The fist displayed equivalence is part of a weak equivalence

$$ B(CRS(\Pi X_*, C )) \to (BC) ^X$$

which does give information on homotopies in this case. See the above mentioned book, Theorem 11.4.19.