The core does not change its rotation rate as the magnetic field changes. The dynamo like affect is an chaotic system which produces not just a single north and south magnetic field but produces a number of north and south regions in the outer core. The south magnetic pole, which is at the north end of the rotation axis at present, and the north magnetic pole which is at present at the south end of the rotational axis are an averaging of many north and south regions in the outer core. As the north and south pointing regions redistribute in the outer core, the measured magnetic field will gain or lose strength as well as wander around as it always has done. The reversal of the magnetic poles is not a migration of the poles to opposite ends of the rotation axis but just a redistribution of many north-south regions in the outer core, but being chaotic it is difficult to predict instabilities in the net field will occur.
The rotation rate on the north south axis can be changed by redistributing mass, adding mass from outside of the Earth, by gravitational interactions by the Sun, planets, Moon, etc. It is not the magnetic field that produces a change in rotation rate but perhaps the rotation rate changing the production of the magnetic fields of the Earth.
By the way, is there any reference that Ben has ever completed a BS or more in any hard science or math? If not, his would not be the bandwagon to jump onto.
There is some true stuff and some intermixed non-science mixed in with his quasi-conspiracy ideas.