It sounds like you found your problem but one thing to note and may be worth checking while you have it apart, just cause you read 12v at the connector does not mean it is good voltage that can carry enough amperage to spin the motor. There are a couple splice connections in the harness in the tailgate that are sealed with electrical tape from the factory. I found mine were all green and corroded and had to cut and resplice those connections to get proper power supply to the motor.
Also note the motor grounds through the tailgate hinges. If the hinges are rusty this can hinder the operation similar to the power supply side (bad ground). A simple fix for the ground is running a ground wire from the gate to the body or frame.
Also note the motor grounds through the tailgate hinges. If the hinges are rusty this can hinder the operation similar to the power supply side (bad ground). A simple fix for the ground is running a ground wire from the gate to the body or frame.

