I know what friction is, thankyouverymuch.
Would you now like to go ahead and argue that when there is no slipping (motion) between your clutch plates in your car while driving, there's no friction between them?
I don't think the string is sliding around the diabolo. I do think there's friction between the diabolo and string, and the string wrapped next to each other.
I also think that the diabolo as a whole takes pressure of the gripper by distributing tension to itself and it's mounting point.