I think this can be made into a proof without being circular. In "Configuration spaces and iterated loop spaces," Segal proves that the group completion of the monoid of configurations of distinct unordered points in $\mathbb{R}^n$ is $\Omega^n S^n$. $\pi_0$ of this monoid is the natural numbers so its Grothendieck group is the integers. This gives $\pi_k(S^k)=\mathbb{Z}$. Since the abelianization of the braid groups is the integers, we have that $\pi_3(S^2)=\mathbb{Z}$. Since the abelianization of the symmetric groups is $\mathbb{Z}/2$, we have that $\pi_{k+1}(S^k)=\mathbb{Z}/2$ for k>2.