It has been close to 20 years ago now, but when I was in my Automatic trans class at the trade school I attended we built 4L60Es. There were like 20 of us in the class and we built like 5 units in groups of 4. If I remember correctly the problem is actually more low line pressure causing lack of cooler flow. If the bushings, lip seals, etc are worn and have fluid leak, the converter charge and thus cooler flow are limited. A 700r4 and 4L60E do not have much in the way of excess pump flow especially at lower rpm. When we had them hooked up to gauges on the transmission dyno the the cooler line pressure was extremely low even on units that had been gone through while they were at low rpm. When you shifted it from say Park to Drive the cooler line pressure was non existent during the shifts. One of the units had Zero cooler line preasure and low line pressure. Subsequent tear down on that unit found that a teflon sealing ring had been pinched during assembly. It still shifted through the gears but would not apply lockup and would have failed in a short time in a vehicle. That team had to go back through that transmission, find and correct the issue before they had a passing grade for that transmission build.
Similarly the AOD that I had a hand in building would not shift. On the dyno we noticed it did not have any governor pressure to the governor pressure tap. We removed the tail housing, swapped the governor and then it shifted fine. When we pulled the old governor apart it had rusted stuck valving inside. Sticking governors were a problem on the old AODs.