Recall the Hopf fibration $\eta\colon S^3 \to S^2$, which is the generator of $\pi_3S^2 \cong \mathbb{Z}$. Also recall the
Freudenthal Suspension Theorem.
If $n>k +1$ then the suspension homomorphism $\Sigma\colon \pi_{n+k}S^n \to \pi_{n+k+1}S^{n+1}$ is an isomorphism; if $n=k+1$ then it is surjective.
The suspension homomorphism $\Sigma \colon \pi_3S^2 \to \pi_4S^3$ is therefore a surjection. One can compute by a standard spectral sequence argument that $\pi_4S^3 \cong \mathbb{Z}/2$, where the non-trivial element is therefore given by $\Sigma\eta$. Since $\pi_4S^3$ is in the stable range, it follows that $\Sigma^n\eta$ generates $\pi_{n+3}S^{n+2}$ for all $n\geq 0$ and we could call the corresponding element in the stable stem also $\eta \in \pi_1^s$.
Now consider the composition $S^4 \stackrel{\Sigma \eta}{\to}S^3\stackrel{\eta}{\to}S^2$ which we could write "$\eta^2$". It's easy to see this is not null-homotopic (consider $\pi_4$!) and by more spectral sequence calculations we know $\pi_4S^2\cong \mathbb{Z}/2$ so $\eta^2$ is the generator. Even more spectral sequences show that $\pi_5S^3\cong \mathbb{Z}/2\cong \pi_6S^4$. Comparing these degrees to the suspension theorem we see 1) $\pi_6S^4$ is in the stable range so $\pi_4S^2$ happens to have the stable value, 2) the suspension morphism $\pi_5S^3\to\pi_6S^4$ is a surjection onto the stable value, which is therefore an isomorphism because they are both $\mathbb{Z}/2$, and 3) we get no information about $\pi_4 S^2 \to \pi_5 S^3$ so it's not immediately obvious from the suspension theorem alone whether or not $\eta^2\in\pi_4S^2$ should survive to the stable stem. It is indeed true that $\eta^2$ does survive and therefore the suspension map is an isomorphism in this degree, but I have only found arguments which use relatively sophisticated tools (see below).
Question: Is there a simple "low tech" proof that $\eta^2$ survives to the stable range?
I have to try to be specific. Here by "low tech" I essentially mean anything in Spanier or Hatcher plus a little bit more. For example the suspension theorem and the Serre spectral sequence are obviously fair use since they have been used above to show $\eta$ itself survives to the stable stem. I'm asking for a simple proof because there is typically a trade-off between sophistication of tools and sophistication of details, so I'm wondering if there is a clever elementary argument or if this result just has a significant lower bound in terms of complexity.
Here are examples of what would be considered "high tech" from this point of view:
Use the $2$-local EHP sequence of I.M. James to study $\Sigma\colon \pi_4 S^2 \to \pi_5 S^3$. The values of the relevant groups have been given above (other than $\pi_5S^5\cong \mathbb{Z}$) and are derivable by Serre spectral sequence in a standard Postnikov tower argument. You find that the suspension map in question is an isomorphism, and therefore $\eta^2$ survives in the stable stem. I consider this to be "high tech" for these purposes because the EHP sequence is an amalgamation of relatively sophisticated concepts including Whitehead products, the Hopf construction, and localization. If an argument only used Whitehead products for example that would be considered lower-tech than the full EHP, since there is at least an exercise set on Whitehead products in Spanier.
Compute the $E_2$ page of the mod $2$ Adams spectral sequence, for example using the May spectral sequence in degrees $< 14$. You find an element $h_1\in Ext^{1,2}$ whose square $h_1^2 \in Ext^{2,4}$ is non-zero, and they're both permanent cycles. This spectral sequence converges like $E_*^{p,q}\Rightarrow \pi^s_{q-p}$ so we know $h_1$ has to represent $\eta\in \pi_1^s$ and that by multiplicativity $h_1^2$ has to represent the square of $\eta$, therefore $\eta^2\neq0$ in $\pi_2^s$. This argument is obviously MUCH more sophisticated than EHP, but with the same amount of work you can also see $\eta^3\neq 0$.
There is also a way of using secondary cohomology operations, which is referenced in this StackExchange question, apparently contained in Harper's book. I haven't looked at this argument yet, but I will draw a line and say primary operations are ok, secondary operations are over the line. That being said my understanding is that $\eta^2$cannot be detected by primary operations.
I hope my question is not still too vague. I think there are many interesting avenues for establishing this result, so if you know a good argument that might not be low tech enough I would still like to see it. I may find that the true answer to my question will be something like "The EHP sequence is not actually that high-tech", and that asking for a proof using Spanier alone might not be reasonable.