The steering twitch is a dynamic balance problem, possibly amplified by a little wear in king pin bushings, or tie rod looseness. On the rear, the tires are almost certainly the problem. Have the tire shop spin up an empty wheel, and see if it has excessive runout and/or balance problems. Mud & snow tires are notorious for having heavy spots, and replacing the tire is the only solution to the problem. It sounds like your tire shop is trying to weasel out of replacing the defective tires.
Jerry

"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
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