The discussion linked to in the arstechnia.com article is fascinating. The paper's author has blinders on.
https://pubpeer.com/publications/3418816F1BA55AFB7A2E6A44847...
Most of the wobble would be Jupiter, so shouldn't Jupiter's orbit be the approximate period of the wobble? Which is 11 years, which is obviously way too short to be in a discussion about global warming.
I found a picture on Wikipedia that seems to confirm that the barycenter takes about a decade to go from near the center of the sun to roughly the furthest away that it gets.
https://upload.wikimedia.org/wikipedia/commons/0/0c/Solar_sy...
So, there is a definite, well-known, historically traced effect that occurs on around that time scale. It would take serious study to move past the coincidence level, but it's not something that can be dismissed out of hand, either.