Studying higher categories from the bottom up

Well, what is an $\infty$-category? Most people who study $\infty$-categories would, if pressed for an actual definition, say something like "a quasicategory" or "a complete Segal space". But any such definition consists of a collection of sets and functions between them satisfying certain axioms. So in that sense, we always study the more complicated $\infty$-categories using the simpler sets. Even homotopy type theory, which treats $\infty$-groupoids or $\infty$-categories as "synthetic" objects that aren't built out of sets, uses set-like structures at the meta-level, with definitional equalities between terms.

Quillen model categories, and related tools like fibration categories or cofibrant replacement comonads, are another way of studying $\infty$-categories using 1-categories. The Riehl-Verity followers who say "$\infty$-category" to mean "object of some $\infty$-cosmos" are using this approach, since an $\infty$-cosmos is a certain kind of fibration category. Homotopy categories and homotopy 2-categories have already been mentioned as another way to study higher-categorical things with lower-categorical ones; "derivators" are an enhanced sort of homotopy category that's especially good for this purpose. (I wrote a blog post about some of these approaches.)