In those days we were taught to apply kotegaeshi without turning the wrist to the outside...which requires that the uke take a hard fall over the arm.....rather hand was wrapped into the wrist on the thumb side and slightly to the outside....and a drop and down. I have gone back to this as it is to me much more effective and causes not direct injury to the uke while getting them on the mat. You destabilize them, drop (slight) and imbalance...they are on the mat without an opportunity to recover.....the connection is through the wrist..elbow..shoulder and center.
Actually that way of doing kotegaeshi (without the wrist turn to the outside and more dangerous ukemi) became our "standard form" of kotegaeshi when Kobayashi broke from Tohei and formed Seidokan. I haven't had to take that hard fall in a very long time. I appreciate this more and more as I get older...