Recall that Kan's first paper was cubical, but subsequent papers were simplicial as difficulties were found with the standard cubical category, in particular that cubical groups were not Kan, and the realisation of the cartesian product of cubical sets did not have the homotopy type of the product of the realisations. These deficiencies have been overcome by using cubical sets with connections (as developed in work with Higgins) in papers by A. Tonks, (JPAA 81}~(1) (1992) 83--87) and G. Maltsiniotis, (HHA 11~(2) (2009) 309--326.) respectively.

In my own work with Higgins, we have used cubical sets extensively see the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids. One major reason for the success of cubical methods is that they can easily express multiple compositions:

Thus in the above diagram, you can ask non mathematicians if they think that there should be some mathematics in which the left hand square can be regarded as some kind of composition of the right hand square, and they will happily agree. Fortunately, this language exists, as double categories, and in the case of the cubical singular complex we can also write down such compositions as a composition $[a_{ij}]$ of an array $(a_{ij})$ of cubes. (See Remark 13.1.11 of the above book.) So we have an "algebraic inverse to subdivision" and this can be used to obtain new local-to-global results. The notion of "connections" on cubical sets arose in trying to define "commutative cubes".

The main theorems of the book were envisaged cubically, and eventually realised, but I feel that simplicially or globularly they would not even be thought of. Indeed, this programme was, what should I say, not welcomed by some.

Further, cubes are excellent for homotopies and higher homotopies, because of the rule $I^m \times I^n \cong I^{m+n}$.

So my answer to the question on $\pi_2 X$ and how to see it simplicially is that one should look to the geometry and not try to force an intuition into an uncomfortable mode, especially when a convenient and natural mode is available, or at any rate, the pros and cons should be thought about carefully.

See also my answer to Generalized Categories for "Higher Homotopy Groupoids" .

Another recent reference is Modelling and Computing Homotopy Types:I.

innerKan complex, also known as a quasicategory, at least when $X$ is a simplicialset. $\endgroup$ – Zhen Lin Apr 26 '13 at 17:48globular set. $\endgroup$ – Hurkyl Dec 20 '17 at 9:58