Past exam of the mathematical methods II course of the University of Oxford / 2013 / 9 / d

There is no contradiction. The domain on which the field is smooth is the punctured plane, which is not simply connected. A loop enclosing the origin cannot be spanned by a planar region entirely inside that domain, so the smoothness hypothesis needed to apply Green's theorem to its full interior fails. A loop that bounds a region avoiding the origin does have zero circulation.

Locally the field is the gradient of the polar angle, but that angle cannot be made a single-valued continuous potential on the whole punctured plane. The nonzero period $2\pi$ around the hole measures this global obstruction, despite the field having zero local curl.