Does the Commutator of a brushed DC motor cause inductive kickback
Yes, a mechanical commutator causes inductive kickback when switching off the current in a winding.

However, this kickback isn't in a place where it can hurt your circuit.  Think about it.  If the kickback results in high enough voltage to arc across the contacts (which almost certainly happens when the contacts initially separate), then from the circuit's point of view the contacts are still closed, maybe just with some extra resistance in series.

The problem with inductive kickback due to mechanical commutation isn't damage to your circuit, but lots of nasty voltage spikes that radiate interference.  This is a nasty problem because it's hard to add snubbers in the right place.  These would have to be across the rotating coils.  That is done sometimes, but is not a easy mechanical problem due to the potentially high forces from the rotation.