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 Blackers-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.

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.

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.

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 Blackers-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.

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 Blackers-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.