If the OP is taking a physics class, she will be learning to work with actual formulas. You are right, generally we ignore calculations involving the barycenter when mass is significantly larger than the other. In this caseypu can work with centripetal force.
When the masses are equivalent, this simplification isn't valid. In the case of the earth and sun the masses are very different. Generally in beginning physics classes we use the simpler equations.
The more important point is the conflict between mathematics and intuition. Very often the mathematics of physics are different than the intuitive answer. In science when this happens intuition must be discarded.