It was George W. Whitehead who first gave a correct calculation of the $2$-stem $\pi_2^s$ in The (n+2)nd Homotopy Group of the n-Sphere. The problem comes down to determining the suspension homomorphism $E\colon\pi_4(S^2)\rightarrow\pi_5(S^3)$, which is known to be an epimorphism with domain cyclic of order $2$ generated by $\eta\circ\Sigma\eta$ (here, $\eta$ is the Hopf fibration). The central tool in these investigation are "generalized Hopf invariants" (GHI) $H$ introduced earlier by Whitehead in A Generalization of the Hopf Invariant. The crux of Whitehead's argument is to establish that, in the metastable range $q\le3n-2$, if $\alpha\in\pi_q(S^n)$ satisfies $E(\alpha)=0$, then $H(\alpha)$ is $0$ or $2$-divisible depending on the parity of $n$ (in fact, the former claim holds in the larger range $q\le4n-4$). The rest is relatively straightforward.
This appears to highlight a relationship between the $E$ and $H$ morphism in the EHP sequence (granted, I don't know for sure the GHI is equivalent to the $H$ map in the sequence, but I'll save that worry for another time), which is not just captured by its exactness. This is proven algebraically by setting up a commutative diagram involving the "Freudenthal invariants" $\Lambda_0^{\prime},\Lambda_0^{\prime\prime}$. Whitehead defines these entirely explicitly, yet this so explicit that I'm not really seeing the "bigger picture". My questions are: Is there a modern account of this method of proof? (Modern textbooks typically use spectral sequence approaches and even Whitehead's own 1978 book Elements of Homotopy Theory rather uses Steenrod squares.) Is there a conceptual explanation of these Freudenthal invariants? (There seems to be little writing on them.) Is there a "bigger picture" explanation of the relationship between $E$ and $H$ at the heart of this proof in terms of modern machinery (e.g. the EHP spectral sequence)?