You're welcome. It's a general principle in physics that we have to be careful with our mathematical models lest we read too much into them. Concepts like center of mass, quantum wave functions and the like are useful because they make it easier to predict the outcome of an experiment in the real world, but that's not always the same thing as providing insight into what's "really" going on.
When Newton devised the center of mass concept, he showed that if you divide up a large, massive object like a planet into a whole bunch of little objects stuck together, each with a mass and its own position, and then compute and add up all the gravitational forces on some distant object from these teeny little objects, you get the same answer as a far simpler calculation: pretending that all of those little masses are piled on top of each other at a single, tiny point in space corresponding to the real object's center of mass. It means that a black hole and an ordinary star with the same mass will have exactly the same gravitational effect on some planet orbiting it, but it does not mean that the star and a black hole are the same type of object!
In fact, you also have to be careful with more ordinary uses of the center of mass. Newton's result is strictly true only for a spherically symmetric object. Real planets are usually not symmetric, with fast-rotating objects like Jupiter (or even the Earth) very decidedly so. If you ignore the planet's true shape and just model it as a point mass, you'll get almost the correct orbital path for a satellite but not quite right. You'll see "perturbations" from classical 2-body motion. To get the right answer you have to take the planet's true, detailed gravity field into account because it changes depending on where you are, not just how far you are.
