Edit: sorry, there used to be a completely wrong solution here (I thought that a certain singular curve was a projective line). Now it is fixed.

There is another solution using algebraic geometry. Identify the ellipse with the projective line by sending the two points where the line through the foci meets the ellipse to $0$ and $\infty$. The map we get by starting from a point on the ellipse, getting the second intersection of the line through it and $F$ with the ellipse, then getting the second intersection of the line through that point and $F'$ with the ellipse is an invertible algebraic map sending $0$ to $0$ and $\infty$ to $\infty$, so it must have the form $x \mapsto cx$ for some constant $c$ (depending on the eccentricity). Thus, the billiard ball approaches the horizontal line at an exponential rate.