This is a very imprecise, general question. Mostly because I'm not exactly sure what I'm after. I just think that I miss something crucial here.
In the context of model categories, homotopy theory, derived geometry and operads ect., it is often a major step to replace a structure, an object or something with a (co)fibrant replacement.
What do we achieve by this? Why is it "better" to work with these replacements, once we add any notion of "homotopy" into our theory? (I know there is no precise meaning of the word "better" in this context, I just don't see the reason, why people put so much thought into these replacements).
Cofibrant replacements are better because left Quillen functors preserve weak equivalences between cofibrant objects, but not weak equivalences in general. Thus cofibrant replacements can be used to derive left Quillen functor, i.e., extract an ∞-functor from an ordinary functor.
See Dwyer–Hirschhorn–Kan–Smith's “Homotopy limit functors on model categories and homotopical categories” for an overview of the modern theory of derived functors.
Relative categories and relative functors form one model for (∞,1)-categories. Specifically, they form a model category that is Quillen equivalent to other model categories: quasicategories (i.e., simplicial sets with the Joyal model structure) and Rezk's complete Segal spaces. This is shown in Barwick–Kan's “Relative categories: another model for the homotopy theory of homotopy theories” and Joyal–Tierney's “Quasi-categories vs Segal spaces”.