Why would there be a need to flush the tranny cooler? Isn’t it just a coiled tube in the radiator tank?
Yet another good question.
The days of my father's cars, whereas the transmission oil cooler was just a simple cooling pipe embedded into the left or right side tank of the radiator are LONG gone. Over 30 years ago, the powertrain engineers sought out better, lighter, more efficient cooling AND heating of the transmission fluid.
To that end we now have coolers which are technically "inter-coolers", by use of passing two liquids across each other in separate pipes.
The cooler is mounted on the side (unsure if on the engine or trans but it doesn't matter) of the trans. Coolant is passed through one set of pipes and the ATF is passed through the other. Think of a common radiator albeit, instead of air and coolant, the media is two different liquids. About three times more effective for it's given size.
Now we take the coolant, which warms up about three to four times faster than the heavier and more viscous oil. We now heat up the ATF, when it's cold so that it works better, and provides more efficient operation sooner. Then after crossing that delta, we use the coolant to control the ATF and keep it somewhere near the 100°C delta.
If you were to cut one of the coolers in half with a bandsaw, you would see some very tight and multiple turn pipe arrangement. The returns (180° turns at the ends) slow down the ATF. That's where any dirt/debris get trapped.
Hence the need for flushing. I'm confident that the service staff at Hyundai/Kia have borne this out hence the "recommend" on the servicing chart.
Me, I change the fluid and filter sooner (remember that almost all ATF is VERY high detergent) so that we can negate some of the dirt/debris in the cooler passages.
HTH?