Yes, the 5V supply should be regulated to a nice steady voltage of approximately 5V.
The obvious solution is to contact your supplier and explain the fault to them. If they agree that the controller is faulty, they should send you a replacement controller to put the fault right under warranty.
If you would prefer to use your external controller, you will need to make up a whole new motor cable to extend the leads from the stator to the controller using three thick wires and five thin ones as shown here:
Then you would need to modify all the wires from the harness to enable them to connect to the external controller.
I would strongly advise you
not to attempt the controller conversion, unless you fully understand what is involved, and have the necessary skills, wiring materials and equipment etc. to complete this modification correctly.
Alan