Mike Shulman, in his article Homotopy limits and colimits and enriched homotopy theory observes (towards the end of page 8) that it may be the case that a functor $F: C \to D$ between homotopical categories has a (say left) total derived functor $\mathbf{L}F : \mathrm{Ho} C \to \mathrm{Ho} D$ but not a point-set derived functor. He also mentions that counterexamples are set in model categories whose factorisations can not be made functorial.
More concretely, what is a counterexample to this fact? Does this come from a general construction? Meaning, can such a counterexample be found in every model category with non-functorial factorisations?
The only idea I have is to take $F$ to be the identity of such a model category. In this case one may construct a total derived functor as explained by Dwyer and Spalinski, since, even if there is in general no cofibrant replacement functor $Q: C \to C$ there is at least a cofibrant replacement functor defined on the homotopy category. But is this enough? I am not completely sure that this can not be lifted for sure to the category $C$. So this, may be related to the following question: is it possible to have a model category whose factorisations can not be made functorial, but nevertheless endowed with a cofibrant replacement functor?
