Quillen's original proof (in Homotopical Algebra, LNM 43, Springer, 1967) is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the classifying space of a simplicial group is a Kan complex. This proof has been rewritten several times in the literature: at the end of
S.I. Gelfand and Yu. I. Manin, Methods of Homological Algebra, Springer, 1996
as well as in
A. Joyal and M. Tierney An introduction to simplicial homotopy theory
(I like Joyal and Tierney's reformulation a lot). However, Quillen wrote in his seminal Lecture Notes that he knew another proof of the existence of the model structure on simplicial sets, using Kan's $Ex^\infty$ functor (but does not give any more hints).
A proof (in fact two variants of it) using Kan's $Ex^\infty$ functor is given in my Astérisque 308: the fun part is not that much about the existence of model structure, but to prove that the fibrations are precisely the Kan fibrations (and also to prove all the good properties of $Ex^\infty$ without using topological spaces); for two different proofs of this fact using $Ex^\infty$, see Prop. 2.1.41 as well as Scholium 2.3.21 for an alternative). For the rest, everything was already in the book of Gabriel and Zisman, for instance.
Finally, I would even add that, in Quillen's original paper, the model structure on topological spaces in obtained by transfer from the model structure on simplicial sets. And that is indeed a rather natural way to proceed.