The cofibration/fibration $\leftrightarrow$ epi/mono confusion

This is probably not a full answer to your question, but I think it is a remark worth to make:

It is actually a couple of remarks:

1)If you have a weak factorization system where either the left class is made of epis or the right class is made of monos then you immediately get a unique (orthogonal) factorization system: any pair of diagonal filler in a lifting square will be either equalized by the mono on the right or co-equalized by the epi on the left.

2) model structure where one of the weak factorizations are unique/orthogonal factorization system are not the interesting ones. They corresponds to trivial model structures on reflective/co-reflective subcategories.

So It makes sense that the model structure we study in practice are "as far as possible" of satisfying the first condition. and this tends to means that cofibration will be monos and fibration will be epis (but it is not always the case).

Also notes that it follows from the axiom of model categories that trivial fibrations with cofibrant target are split epis and trivial cofibrations with fibrant domain are split monos, so unless the only weak equivalences are the isos you always gets that some cofibrations are non trivial monos and some trivial fibrations are non trivial epis.

Edit: so to sum up, this is closely related with Chris comment: it boils down to the difference between unique factorization system and weak factorization systems.

The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the homotopy theory of (-1)-truncated sets. One can also define a similar model structure on simplicial sets, where the weak equivalences are the maps which are surjective on pi_0 (i.e, homotopy epis) and whose fibrations are the Kan fibrations whose homotopy fibers are all (-1)-truncated (i.e., homotopy monos). This is a also a model for (-1)-truncated spaces/sets. A similar thing can be done in others contexts as well, for example, if you can replace simplicial sets by a model category of simplicial presheaves which presents a certain $\infty$-topos, then you can often left Bousfield localize $M$ such that the new weak equivalences are homotopy epis and the fibrations (at least between old fibrant objects) are the old fibrations which are also homotopy monos. The new model category will present the homotopy theory of (-1)-truncated objects in the original $\infty$-topos. I will go on a limb here and suggest that the (epi,mono) type factorization systems, which appear very typical from a 1-categorical point of view, are, from a model categorical point of view, essentially the particular case of (certain kinds of) (-1)-truncated homotopy theories.