Other way around, the further apart the objects are the less likely the barycentre is to be inside one of them, you can picture it as a rubber band with a dot drawn on it, the more you stretch it the further the dot gets from both ends even if it gets further from one end faster.
Yeah, I went and checked after posting.
My hunch is that if the moon was closer it would ‘drag’ the barycentre closer to the moon.
Which, given the moon is slowly receeding, means it was probably a binary early on in the formation of the solar system.
Other way around, the further apart the objects are the less likely the barycentre is to be inside one of them, you can picture it as a rubber band with a dot drawn on it, the more you stretch it the further the dot gets from both ends even if it gets further from one end faster.