As to the average position, I would expect it to be the center of mass. The equation of motion is decomposable into two one-dimensional equations in orthogonal coordinates, reducing the question to "what is the average of a sinusoid".
In this view ("scaffolding", to borrow Henry's term from another recent email) the ellipse is not the fundamental solution. The ellipse is assembled from two more basic bits: two one-dimensional sinusoids.
Can you give an example of this decomposition that isn't a circle? The straightforward decomposition would be x = A sin t y = B cos t but the variable t here is not time, since the planet travels faster at perihelion than aphelion, and these equations do not reflect this. The average position of the planet averaged over t would be the origin, the point midway between the foci, but if we average over time instead, we'll get a different point, and I don't see a way to do the decomposition in a way that makes this calculation easy. The center of mass can't be the right answer. In the limit where the planet has tiny mass compared to the sun, symmetry tells us that the average position lies on the major axis. But since the planet moves slower on the side away from the sun, the average will be biased in that direction, so it will be on the other side of the center of the ellipse from the sun. Andy