Not everyone applies kotegaishi in this manner. Some practitioners treat it more in a manner which I would call kokyunage, where they are drawing uke's wrist away from their center, then about/above shoulder height, then out away and down. Hard to explain, and rough to fall from.
To me this still sounds like what I said, using the hand to move the body, via the other joints in the arm. It doesn't sound like it should be hampered by a stiff wrist. ?
I sprained my thumb once and spent a month or so in a wrist splint, and a lot of people had no trouble doing kotegaeshi on me - more than one different kind of kotegaeshi, with some requiring big breakfalls and others going more straight down. As long as they were trying to move me and not just make me go 'ow', it worked fine.