Consider two examples:
1) US708 is traveling at about 1200kps. For simplicity, lets give it one solar mass and have it travel near us at 15000 a.u. In one year it would move about 1 deg. The barycenter between it and the Sun would be half-way between them. Would the Sun gain a new orbital vector around this encroaching star (and barycenter)? I've been thinking it would; simple moment math. But the answer is more likely, no, that gravity is the proper model. [Of course, it will gain a vector, but I am meaning a very strong tangential one matching that of US708 in a binary-type motion for that period, even though orbital acceleration must be considered, too.]
2) Saturn's Epimetheus and Janus exchange orbits close to every 4 years. As they near each other, excluding Saturn for a second, their mutual barycenter is easily determined, if we know their mass and locations, and we do. In a heavy-thinking (bull-headed) leverage model, the faster inner moon should swing past the slower moon and outward enough to become the outer orbiting moon. Vice-versa for the other moon. This is exactly what doesn't happen. And I don't think the kinematics improve by adding Saturn to this dynamic. The gravity/Kepler model, however, does work.
Adding the Sun would only change how we plot the motions, which is where the barycenter comes in handy, if it is offset from the c.g. of the primary body. That's the point (no real difference) you both are making, I'm sure. A distant quasar to the solar system, or Milky Way, gives us only a superfluous barycenter; it's useless because there's no effective relative orbital motions. The car/Earth barycenter is also superfluous since the c.g. of Earth itself suffices. [There are exceptions -- my original hand-me-down was an old Old's 98, thus moving the Earth/car barycenter twice that of most others.
] The only purpose in giving the car wings is to get orbital motion's back into the picture for a barycenter term.