So if the elevator from the moon extends toward earth and is held in place by Earth's gravity... why stop at geostationary orbit height?[1] Wouldn't you want to

So if the elevator from the moon extends toward earth and is held in place by Earth's gravity... why stop at geostationary orbit height?[1] Wouldn't you want to extend the elevator as close to the earth as you could go? It seems to me that the closer to the earth the elevator goes, the more Earth's gravity keeps its location stable, and the easier it is to get from Earth's surface to the tip of the elevator. It seems to me you'd want to stop the elevator just shy of the strength limitations of the cable material (with some margin of error). [1] I suspect the article may just be wrong about this, since as others have pointed out, the moon has an elliptical orbit that is not geostationary.