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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 ...
Here's a sketch of the proof why realizations of simplicial sets are CW complexes. Roughly speaking, a CW complex is a space built by one-by-one gluing in new simplices along their boundaries. If $X_\bullet$ is a simplicial set, then $|X_\bullet|$ is built by gluing together the spaces $X_n\times\Delta^n$, where you give $X_n$ the discrete topology. But ...