Exccept for either
two places that are on the equator (both great circle and rhumb line),
two places that are on the same meridian or one meridian and it's antemeridian (eg 10W and 170E) (both great circle and rhumbline),
two places that are at diametrically opposite points (eg N pole and S pole) (can be joined by infinite number of great circles which are also rhumb lines)
then the direction of a rhumb line joining those points and the direction of a great circle joining those points will be different except at the point half way between them.
At the departure point or destination the direction of the great circle will always be closer to the nearest pole than the direction of the rhumb line.
So since they are in different directions, starting in the direction of the great circle and flying a rhumb line will cause the aircraft to not only fail to arrive at the intended destination but to also not follow the great circle.