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Most cars do not have a transmission oil cooler. What they do have is an oil warmer really. The bottom part of the radiator on auto equipped cars is usually used as an oil cooler/warmer. These generally are not very effective.
How hot do you think the cold tank on the radiator actually is? Check it. It must be significantly colder than the temperature of the coolant leaving the engine and entering the radiator. In cool weather it will be quite cool. And they are effective--modern transmission fluid is often good for 100,000 miles and the transmission good for 300,000...I know, many exceptions, but these get their job done. My turbo Volvo had the engine oil cooler in the radiator hot tank and the transmission cooler in the rad cold tank. When I sold it at 212,000 miles, the transmission was fine.
Add-on coolers are very good when hauling heavy loads. I put the Dana Long Tru-Cool stacked plate cooler on my Tundra with a valve so I could take it out of the circuit when I wasn't towing. the Tru-Cool has an internal bypass so the cold fluid isn't chilled too much. Transmission fluid needs to be normally warm, maybe 140-160F