Why does the Earth revolve with the Sun at one of its foci? Does the other focus do nothing? Why is there this asymmetry in our solar system?
I'm almost certain there used to be an answer to this question, but it seems to be gone, I'll write another one. The Earth and Sun both orbit their mutual barycenter (disregarding any other objects of course). That one point is a focal point of both ellipses, and all three focal points are collinear.
It may appear asymmetric because the Sun's motion is so small, which comes from the asymmetry in the masses. Imagine smoothly scaling the Sun down to an Earth mass. As you did so, the ellipses would approach each other in size, and your missing symmetry would be restored. The key is to notice that there are three, not just two, foci in the system.
The other focus fixes the solution: it selects a particular ellips out of the infinite solutions that could revolve around the first focus.
Note that the law of equal areas only depends of the central force, this is the focus where the Sun is.