The barycentre of the Solar System is usually just above the surface of the Sun, in the direction of Jupiter (more or less -- because Saturn also has great influence on the position of the barycentre).
When Jupiter is on one side and the other giants (especially Saturn) are on the other side of the Sun, then the barycentre is inside the Sun's sphere.
The Sun does orbit this barycentre, with a weird orbit made up of the influence of all the other planets - especially Jupiter. If someone from outside was observing our Sun to detect this wobble, they would first detect the primary period of 11.8 years (because of Jupiter) and a secondary period of 29.5 years (Saturn).
Even these two periods would be very tricky to observe (the change in the Sun's orbital velocity is very, very small), and they would hide the other periods caused by the smaller planets.
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In the same manner, Earth and Moon are orbiting each other (the barycentre is inside Earth's sphere), and it is this barycentre that is in the "nice elliptical" orbit around the Sun. Earth and Moon, individually, are zigzagging back and forth along this ellipse, with the Moon zigging when Earth is zagging, and vice-versa (the Moon's zigs being 80 times larger than Earth's zags).
The "nice ellipse" itself is changing slightly from year to year (by roughly 500 km) because of the gravitational interference from the other planets (mostly Jupiter and Saturn).