Algebraic topology and homotopy theory with simplicial sets instead of topological spaces To quote Kerodon:

In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces.

A similar quote can be found in the mathscinet review for Kan's On c. s. s. complexes:

In recent years it has become evident that for most purposes in homotopy theory it is more convenient to use semi-simplicial complexes instead of topological spaces.

For instance, I know that to special simplicial sets called Kan complexes one can assign higher homotopy groups and prove and analogue of Whitehead's theorem. This certainly demonstrates that one can do some homotopy theory with simplicial sets / Kan complexes.
If I open an introductory book on algebraic topology or homotopy theory (such as Hatcher's), do all the main theorems admit analogues in the world of simplicial sets or Kan complexes (replacing topological spaces)?
I'd be totally happy if you could give me, say, four theorems in algebraic topology / homotopy theory that can be phrased for simplicial sets, together with the original reference. I'd also be interested in whether these theorems are more algebraic topology or more homotopy theory (I don't really know the difference).
 A: It certainly used to be the case that if you went to an algebraic topology research talk, when the speaker said, "Let $X$ be a space," then there was about a 50% chance that they really meant "Let $X$ be a simplicial set": simplicial sets are that intertwined with homotopy theory.
More precisely, Quillen's model category framework allows you to start with a category $C$, say simplicial sets or topological spaces, add some extra structure, and then define what is called the "associated homotopy category," $\text{Ho}\,C$. Quillen showed further that the homotopy category for simplicial sets is equivalent to the homotopy category for topological spaces, and therefore if you want to study homotopy theory, you can use either topological spaces (with CW complexes as a distinguished subcategory) or simplicial sets (with Kan complexes as a distinguished subcategory) interchangeably. Even better, you can switch back and forth depending on which setting is more convenient for proving the result you happen to be interested in at the time.
A: It depends on what you mean by "all results". Of course results regarding manifolds or vector bundles do not admit statements completely internal to the world of simplicial sets (although most of them are just an application of $\operatorname{Sing}$ away from the world of simplicial sets).
But if one concentrates oneself to the "purely homotopical" statements (like, say, the Freudenthal suspension theorem, the Whitehead theorem, the Brown representability theorem and the Blakers-Massey theorem) they can all be stated in terms of simplicial sets (or, better, Kan complexes).
Indeed there is a textbook by Goerss and Jardine that does most elementary homotopy theory in terms of simplicial sets.
