Proving that the category of simplicial sets carries the Quillen model structure is undoubtedly difficult; the book by May and Ponto "A more concise course in algebraic topology" makes a considerable effort in giving an elementary view of this proof: section 17.5 begins with
I [Peter May] have long believed that simpler proofs should be possible. Correspondence between him and Pete Bousfield in the course of writing this book have led to several variant proofs, primarily due to Bousfield, that are simpler than those to be found in the literature.
Near the end of the chapter, there is a paragraph trying to motivate this intrinsic difficulty:
Proposition 17.5.12 means that the left adjoint T [geometric realization] creates the weak equivalences in sSet. In most other adjoint pair situations, it is the right adjoint that creates the weak equivalences. The difference is central to the relative difficulty in proving the model category axioms in sSet.
It caught my eye. I would like to have a precise explanation for this fact. Why should such a creation property be involved in the difficulty for $\mathbf{sSet}_\text{Quillen}$ to be found?