Im not sure how they could tell the differential was bad just by driving it without taking it apart and checking. I would have them show you what was bad on the old one. The bearing would be the rear wheel bearing. The tire spinning could have been your old tires. Did it still happen with the new tires? Now for the confusing part. There is no skid plate on the rear differential. Are you talking about the front skid plate or maybe gas tank skid plate? Theres no reason to have removed those to do the front brakes or rear axle and differential.
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