 Solution

At an area critical point the assumed first-variation identity holds for every smooth $\phi$ compactly supported where ${\kappa }_g\ne 0$:The fundamental lemma of the calculus of variations gives
$({\kappa }_g^{-1}{)}^{\prime }=0$ on each component of $\{{\kappa }_g\ne 0\}$. Thus ${\kappa }_g$ is constant there. A nonzero constant component cannot have a boundary point where continuity makes ${\kappa }_g=0$. Since the closed curve is connected, either ${\kappa }_g\equiv 0$ or it is one nonzero constant everywhere. Therefore every area-maximizing curve has constant geodesic curvature.