I've never seen such an interesting analysis problem :) Here is my current state, which I post because I like this form in itself and because I hope somebody finishes it.
We can reformulate what we want to prove as follows, by taking the direction of $F'(t)$:
Consider a smooth curve on a sphere $S^{n-1}$ that doesn't lie inside any half-sphere. Prove that there exists an equator that intersects the curve in at least $n$ points.
(Note that an there is $n-1$-dimensional family of equators.)
Now suppose there were a contradictory curve. I'm currently thinking about shrinking it: if we can do it keeping the offending property properties (using that there are no $n$-intersecting equators) then we'll arrive to contradiction since once the curve is small it's definitely inside half-circle.
