Mihawk never stated he never defeated Shanks. That is headcannon so stop.
Mihawk just no longer considers Shanks a worthy opponent since he lost his arm.
Which means Mihawk did find Shanks to be a worthy opponent prior. It doesn't mean he didn't beat Shanks.
How would Mihawk determine 1 arm Shanks is not a worthy opponent? By fighting 1 arm Shanks.