A famous result due to Serre states that any simply-connected finite CW complex with non-trivial $\mathbb{Z}_2$ homology has infinitely many non-zero homotopy groups. (In fact, Serre proves more than this: the homotopy groups are non-trivial mod 2). The result has since been generalized to apply to any prime $p$.
I'm aware of two distinct proofs of this theorem. The first, due to Serre and extended by Umeda, is somewhat analytic in nature and follows from a careful examination of the Hilber-Poincare power series associated with the cohomology of a space. A second more homotopical proof is given by McGibbon and Neisendorfer and, independently, Zabrodsky which relies on Miller's proof of the Sullivan conjecture and a link between $B\mathbb{Z}_p$-nullification of a space and its $p$-completion.
Both of these proofs involve significantly more machinery than, say, Serre's other quantitative results on homotopy groups (finite generation, finiteness, etc.). The first using analytic tools and the second the substantial assumption of Miller's theorem.
My question is similar in spirit to this one: have any proofs of Serre's result been produced in the intervening years which are more direct in nature? More generally, I am curious if there are known proofs for this result which do not follow the two approaches mentioned above.
