Here is an attempt at a proof of the hairy ball theorem, that I'm not sure is quite right: Suppose we had a non-vanishing continuous tangent vector field on $S^2$. Then it's clear the tangent bundle of the circle would be trivial (you can make consistent coordinate frames everywhere by scaling your vector field to unit length and taking a vector rotated 90 degrees counter-clockwise as your frame), and thus its circle bundle would be trivial as well, being $S^2 \times S^1$, with fundamental group $\mathbb{Z}$. But the circle bundle is acted on transitively and without fixed points by $SO(3)$, so it is homeomorphic to $SO(3)$, which has fundamental group $\mathbb{Z}/2\mathbb{Z}$.
Is there anything subtle I could be missing here? I'm not that practiced with bundles, so I want to be sure. If this does work, how hard is it to generalize to higher dimensions? I think I prefer this proof to others that I have seen: can this argument be found somewhere else?