The Mauser action is considered to be a very precise, very repeatable action, correct? Is that the basis for its claim to accuracy in rifles that use it?

It's considered to be a very reliable and long lasting action. The fit and design keep the action "tight" through repeated use and reduces wear on mating surfaces. A larger-than-most extractor claw reduces the possibility of stuck rounds from breaking the claw or slipping out during extraction. A noticeably smoother action when chambering or extracting a round. I don't know that it contributes to the accuracy of the rifle, would say not in my experience, but opinions vary.

The engineering of these kinds of intricate things is pretty satisfying to behold.