You definitely can over-cool these transmissions. The A750F/760F generate very little heat when running with the TCC locked as there is no slippage occurring and the only heat is from the fluid getting pumped around. My A750F with a Hayden 678 would run as cool as 120F after 30-40 minutes of cruising during below freezing temps outside. Since heat flows from warmer to colder, this means the radiator was actually already warming up the trans fluid - and the Hayden 678 (downstream of the radiator) was then shedding the heat back to the cold air. Had I been running a standalone system as proposed, it would have been running even cooler as there would be less heat going into the external cooler to start with.I don’t think this is the case. With older systems yes but in case the transmission has a thermostat in the circuit. It won’t circulate fluid until the thermostat opens.
This is one of the reasons I’m tuning the fan clutch. You can’t overcool the transmission but sure as hell can over heat it.
Shifting was harsh with this setup, so I made a rubber cover that blocked up about 2/3 of the Hayden 678 for winter use. I am not sure what other problems can be caused by an over-cooled transmission, but it would be running well outside of Toyota's design parameters. IMO they run fine in the 150-190F range, but not any colder. Mine's never gotten much above 200F with a CSF radiator and a Hayden, and that includes lots of towing in 100F+ temps.