The standard definition for simplicial homotopy groups only works for Kan complexes (cf. <http://ncatlab.org/nlab/show/simplicial+homotopy+group>). I learned that the hard way, when I tried to compute a very simple example, i.e. the homotopy group of the boundary of the standard 2-simplex. 
My naive idea to actually compute simplicial homotopy groups for arbitrary simplicial sets was taking the fibrant replacement. But obviously we need a model structure for that. Then again, a weak equivalence in the usual model structure for simplicial sets is precisely a weak equivalence of the geometric realization.(cf. <http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets>) 


As I understand it so far, the only satisfactory way to talk about simplicial homotopy groups requires the notion of "classical" homotopy groups. Hence my question: why does it still make sense, to actually talk about simplicial homotopy groups in the first place?



Why study simplicial homotopy groups?
To compute the homotopy groups of a simplicial set $X$, you need to be able to construct a weak equivalence $X \to Y$ where $Y$ is a Kan complex, and then compute the homotopy groups of $Y$ using the definitions you were discussing.


This might seem circular - you need to detect if $X \to Y$ is an equivalence. However, you can construct $Y$ directly using certain more elementary equivalences. Specifically, for a map of a horn $\Lambda \to X$ we can form the pushout of the diagram $\Delta \leftarrow \Lambda \rightarrow X$, called $X'$; on geometric realizations this is homotopy equivalent because you can construct an explicit retraction. The class of maps $X \to Y$ generated by such pushouts is called the family of anodyne extensions of $X$, and you can always construct an anodyne extension which is a Kan complex by the small object argument (you keep gluing in solutions to every possible horn-filling problem).


If you want a more canonical answer, there is also [Kan's $\operatorname{Ex}^\infty$ construction](https://ncatlab.org/nlab/show/Kan+fibrant+replacement#_functor).


If you want another reference, there are Kan's older papers, and my recollection is that Joyal and Tierney has quite a number of details as well.


