It is not a point. Consider two masses connected by an rope, one in an higher orbit. The entire contraption will move along their common center of mass, and the

It is not a point. Consider two masses connected by an rope, one in an higher orbit. The entire contraption will move along their common center of mass, and there will be tension on the rope. The same thing applies for an lunar elevator, one end is somewhere close to geostationary orbit, the other is the moon, and the entire system moves as a combination of internal tension and the movement of the center of mass.