The Galilean velocity transformation, Eq. 4-32, is so instinctively familiar from everyday experience that it is sometimes claimed to be obviously correct, requiring no proof. Many so-called refutations of relativity theory turn out to be based on this claim. How would you refute someone who made this claim?

In spite the fact that the Galilean transformation appears to be true from the everyday experience one cannot use this experience to refute the relativity theory. When the light was tested against the Galilean transformation, it was observed that light neither appeared to be moving fast or slow relative to another inertia frame of reference. Rather the speed of the light as observed by any observer eventually came out to be the same.
It is important to emphasize that one cannot universalize a principle on the basis that it proves to be true on a common experience because the principle might not work if the experimental conditions are taken out from those common experiences.