The Hopf map is the projection of a circle bundle $h\colon S^3 \to \mathbb{CP}^1 \cong S^2$, where the complex numbers of unit length $S^1 \subset \mathbb{C}$ act on the unit vectors $S^3 \subset \mathbb{C}^2$ by scaling. I am interested in an elementary argument (in particular not using the Hurewicz theorem) showing that this map is essential, or in other words not null-homotopic, and I'm also wondering what were the original techniques to prove this.
I know the Hopf map is essential by considering the long exact sequence of homotopy groups for a fibration. Since $S^1$'s universal cover is contractible $\pi_k(S^1)\cong 0$ for $k > 1$ so there are isomorphisms $\pi_k(h)\colon \pi_k(S^3) \cong \pi_k(S^2)$ for $k > 2$, and since $\pi_3(S^3)\cong \mathbb{Z}$ that means $\pi_3(h)$ is non-zero. However, the only reason I know that $\pi_3(S^3)\cong \mathbb{Z}$ is from the Hurewicz theorem, which is far from elementary. (Note however that this sequence does establish the result that $\pi_2(S^2) \cong \mathbb{Z}$ without using Hurewicz.)
Is there a different approach to showing the Hopf map is essential, which doesn't rely on a computation of $\pi_3(S^3)$? Alternatively, it would suffice to have an elementary way of proving that $\pi_3(S^3) \neq 0$ (without necessarily determining the actual value), since this is the only property I needed to deduce $\pi_3(h)\neq 0$.
More generally I guess I am interested in the history of higher homotopy groups. From what I understand people initially thought that all higher homotopy groups were trivial and I get the impression that the Hopf map was one of the early non-trivial examples. I want to know what were the first non-trivial examples and what were the techniques used, and how much of it depended on difficult results like Hurewicz. (I apologize that this is maybe not the correct forum for math history questions, but I'm not aware of a better one.)