I'm currently trying to understand the basics of homotopy theory for simplicial sets. However, my current sources (Peter Mays "Simplicial objects in algebraic topology" and Kans original "on c.s.s. complexes") are kinda outdated in the sense that, for example, they define the extension condition combinatorically on the set of simplices, instead of via the more modern horn filling condition.
What I'm looking for is a source that uses modern notation, i.e. defines Kan complexes via horn filling conditions, but doesn't take the top down model structure approach (since I want to use this as a motivation on why a model category is defined the way it is).
Have you looked at the book Simplicial homotopy theory by Goerss and Jardine? They have a 60-pages Chapter 1 on simplicial sets, at the end of which they define the model structure on $\mathsf{sSet}$. IMHO the book is quite good. I'm not sure what you mean by "doesn't take the top down model structure approach", but definitions don't fall from the sky and are motivated from topology.