I'd just like to know if proofs of the Hairy Ball Theorem along the following lines are well-known or even somewhere in the literature.

From a given vector field $V_1$ on $S^2$, form another, $V_2$, by rotating each vector 90 degrees. 
Wherever either field has a zero, both must.
From any $x\in S^2$, form the integral curve $\tau_t$ for $V_1$ starting at $\tau_0=x$.
Perhaps for large $t$, $\tau_t$ converges to $y$; then $y$ constitutes a zero of $V_1$.
Regardless, by compactness of $S^2$, sequence $\tau_0, \tau_1, \tau_2,\ldots$ has a convergent subsequence, limit point $z$ say.
Assume now that $\tau_t$ itself does not converge to $z$.
With radius $r$ small enough, a ball $B(z,r)$ will have $V_1$ sufficiently close to constant inside that $\tau_t$ must leave $B(z,r)$ after each sufficiently close approach to $z$.   Accordingly, we can form a simple closed curve $S$ by following $\tau_t$ from some exit of $B(x,r)$, at $b$ say, until its next entrance, say at $a$, finally connecting $a$ and $b$ within $B(x,r)$ with a great circle segment.
$V_2$ flows transversally across $S$, whether inside or outside of $B(x,r)$. 
Since $S$ bounds two (closed) disks, the vector-field version of the Brouwer Fixed Point Theorem now forces a zero of $V_2$
(indeed at least one in each disk if a shared zero doesn't fall on $S$).