Have you looked at the work of Richard Garner and of Emily Riehl?
I'm not an expert on this, but here's what I think I know. Garner has a notion of "algebraic" model category, which I think is rather more than having functorial factorizations. The idea, I believe, is that you know not just whether something is a fibration or cofibration, but why it is. (This builds on work of Marco Grandis and Walter Tholen on "natural weak factorization systems".)
That sounds like it's asking a lot, but Garner has a small object argument implying that any cofibrantly generated model category can be algebraicized. So, for example, this gives you a fibrant replacement monad (I mean a genuine monad, not just up-to-something), a cofibrant replacement comonad, and a distributive law of one over the other.
All I can find about this on Garner's website is this; I suspect he's done more, though. The previous paragraph came from my notes from this talk by Riehl, and there's an associated paper of hers.