The first radiator I did was on my 2006 several years ago. On that one, I replaced the radiator. The second one was on my 2008, which I bypassed, initially, but recently went back and replaced the radiator and re-connected the cooler lines. It wasn't so much that I had an issue with the bypass on the 08, but I got a really good deal on a Koyo radiator for $50 and decided to replace it.
You have the order of the trans cooler system incorrect. The ATF goes through the auxiliary cooler first, then routes through the radiator's integral cooler and back to the transmission. The auxiliary cooler is exactly that: a cooler. The radiator's integral cooler is both a cooler and a warmer and is designed to maintain the transmission fluid within a specific range. Automatic transmission fluid should normally run in the 175-200 degree F range in the transmission oil pan and shouldn't run any cooler than 150 degrees F. With just an air-to-air cooler, or coolers, installed, the temperature could fall below that temperature, especially in very cold weather, which is not ideal for the transmission. Using an air-to-water cooler, like the one inside the radiator lower tank, allows the heat of the coolant to get the temperature of the transmission fluid up to the 150+ degree F range more quickly and allows the ATF to be maintained in the 175-200 degree F range, which is optimal for ATF. This is the reason they route the ATF through the auxiliary cooler, first, then the radiator cooler; even if the ATF is over-cooled when it exits the auxiliary cooler, it will be warmed up and exit the radiator cooler at the optimal temperature. This is also why most vehicles use just the air-to-water cooler from the factory rather than just an air-to-air cooler. Now, if the ATF, as it exits the auxiliary cooler, is too hot (which would be a rare case that would likely only occur under extreme towing conditions), the radiator's cooler would then serve as a cooler to the ATF and air-to-water coolers are far more efficient than air-to-air coolers.
If you are interested, here's a good guide for transmission fluid temperatures: