I guess I was taught different. I install the auxiliary cooler inline after the OEM cooling system. So if there is a heat exchanger in the radiator, the aftermarket unit provides additional heat dissipation after the fluid has passed the OEM cooler.
Installing auxiliary coolers was something that used to be regarded as something that you had to do with off-road vehicles, trucks & vans used for hauling & towing, performance vehicles used for racing, and if the vehicle was in use for hours all day. It used to be believed, and it may have been true, that the OEM system was engineered for short trips, commutes, getting groceries, running errands. Great if you drive for less than 30 minutes to get somewhere, then the car is parked for over an hour to cool off, before you drive another 30 minutes back home.
My pickup is a 2007. When it was new, I bought a large, B&M Racing cooler with a fan. A guy I know who owns a shop told me to hold off on installation. He said to drive it and monitor the temps first, and get a good base of data. This was over 10 years ago, actually 14 years ago, and even back then, he said that the modern cars are engineered better. He told me that modern OEM cooling systems evolved to being efficient, and if my ATF temps spiked to 250, there was a problem with the transmission that a cooler won't fix. Or that I was hauling too much weight, towing too much weight, or just driving way too fast.
I've never overloaded any of my trucks. What I do notice is that if I'm driving 80 - 90 mph, where I go full open throttle to pass slower cars, the transmission is not in overdrive, downshifts heavy, and transmission temps are higher. Or if I am climbing steep grades at freeway speeds, uphill at 60 - 70, the truck does downshift and the engine revs up. Considering that I can't go uphill all day long, or drive 90 mph all day long, the ATF temperature does lower as I slow down, or stop. So in that sense, the cooling system works. With that in mind, I do live in a city with very steep hills, and a lot of rush hour type, stop and go traffic.
So 14 years ago, with the pickup, after driving it around for 1 year stock, I did install the aftermarket cooler. The temperatures still gets up to 220 - 230, and the temps drop back down to 185 - 195. I can't tell if the cooler adds any efficiency towards lowering the temps, as the temps got up to 230 with and without the cooler. I do know that the cooler works. I measure the line temps, and I can see that the line from transmission to radiator is hottest, the line from the radiator is cooler as it goes to the auxiliary cooler, and the transmission line from the auxiliary cooler back to the transmission is the coolest. A thermometer confirms, as does touch of the hand, that the line in side is warm from ATF entering the cooler, and the line out side is cool to the touch. I do know that the fan on the cooler turns on as regulated by a thermostatic switch. But it's impossible for me to guess at how the auxiliary cooler benefitted my truck. No noticeable difference in performance. ATF temperature consistently heats up and cools down depending on driving activity.
The coolant feed to the heat exchanger is controlled by electronic valves. Coolant will only flow as needed. From what I remember, coolant from the overflow tank goes to the heat exchanger, then flows towards the heater core. The heat exchanger has fins for cooling by airflow. I don't know how much cool air you get under the hood, above the transmission case. From my swiss-cheese memory, I have only seen ATF temp from the sensor read up to 220. In theory, 250 is the danger zone where you want to pull over, park, idle the engine, and lift the hood, so that the ATF can cool down. A lot of people, myself included, still aren't comfortable with average ATF temps at 225. I still don't trust that it's fine even at 235. Another thing I don't trust is that when they say the pink color is just a color additive, so when the ATF looks black, it's still perfectly fine.
Only 1 way to know. Test it. I have seen on other forums where people use a point and shoot thermometer to check the aluminum line temperature as the bypass valve closes. Other people have removed the bypass valve, and boiled it to see what temp the water gets to before the valve pops closed. Against conventional wisdom, some people remove the bypass valve entirely, without worry of "overcooling" the transmission. Just a few inches of transmission line will "bridge" the cooling and return lines. Then ATF will always flow through the cooling system. It will take a little longer to "warm up" the ATF. But that does nothing for when the ATF gets really hot. Rare, but it does happen, is the cooler bypass valve being stuck open.