Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa: Are finite spaces a model for finite CW-complexes?
So, when it comes to computing homotopy groups of finitely triangulated spaces, there is really no loss of generality.
One might try to draw as an exercise a finite topological space having the same homotopy groups as the 2-sphere: it will be a set of cardinality $14$, namely the poset (order given by inclusion) of all simplices of the boundary of the $3$-simplex with the order topology.