The Moon also revolves around the center of mass of the solar system. In a sense more than it does around the earth. In particular Moon's orbit around the barycenter is convex - the popular picture of Moon's orbit having loops is wrong in that sense.
My illustrative story from a few years back, was that if Earth were to vanish, the Moon's orbit would merely smooth out a bit, losing a slight ripple. Whereas if Jupiter were to vanish, Io would zoom away, perhaps even leaving the solar system, or impacting the Sun.
