[I re-wrote the question, so Christian's answer becomes obsolete. Sorry for that.]
Consider uncollapsable closed curves on a cone (of reflection) of a given length that pass through a given point and have minimal $\oint_\gamma |\dot{\kappa}|\>ds$ ($\kappa$ = curvature) among all loops of the same length going through the same point.
If the distance of the point to the apex is $\sqrt{2}$ and the given length is $2\pi$, the curve will be a circle.
By which arguments might be told that these curves are exactly the ellipses (as cone sections, which I assume they are) without calculating them?
(How might parabolas and hyperbolas come out as such curves?)
If you mean by "overall curvature" the total curvature of a loop $\gamma$, i.e., the integral $\oint_\gamma \kappa_g\>ds$, then this total curvature is the same for all loops around the apex, by the Gauss'-Bonnet theorem, and can be expressed by the total Gaussian curvature of the cone, which is concentrated at the apex. It seems therefore that all loops are "equators" according to your definition.