The rolling has no physical significance. It doesn't change the intrinsic curvature, which is zero in this example (tidal forces are negligible on small scale). It is true that an actual closed cone has intrinsic curvature, if you include the apex in the parallel transport loop. But you cannot do this here. The diagram is not a closed cone: The cone surface is not closed at the "seam", but continues on a new layer, so you cannot parallel transport around it in a closed loop.