I've just posted an answer on math-stackexhange where the question seems a bit more appropriate.

The infinite dimensional sphere is contractible. And the one point space appears to be another counterexample.

Actually, I remember seeing more unstable charts in the "Oaxtapec proceedings" (CONM146) (Appendix 2 by Paul Shick), but I don't know if this relates to $S^2$. Be warned that it's usually nontrivial to deduce the homotopy from these charts, so this info might not be of much use to you.

You could deduce some information on $\pi_{32}(S^3)$ from Bob Bruners unstable chart on the bottom of math.wayne.edu/~rrb/cohom/index.html (if that's of any help)

If my memory is not mistaken, Mike Hopkins discussed $\pi_4(S^2)$ geometrically in his talk at the Atiyah birthday conference in 2009. The talk can be found here: maths.ed.ac.uk/~aar/atiyah80.htm

I'm using the straightforward approach: the FGL $F(x,y) = x + y + {\mathrm stuff}$ is computed by induction on $deg({\mathrm stuff})$. In each degree one first computes the associativity defect, then adds appropriate correction terms. One final correction term is used to to get the desired $p$-series.