In climates where there very low temperatures, say Minn in winter, the cooler coils in the bottom of the radiator help to warm the transmission oil up. This improves the ability of the transmission to shift smoothly; thick, viscous and cold oil can cause delayed shifts and/or rough shifting. So yes there is a benefit to having the transmission cooler at the bottom of the radiator tank. For most climates though, the oil cooler does just that, cool the oil. Automatic transmissions have a lot of friction material in them and this material generates lots of heat as does oil swirling about in the torque convertor. All this heat has to be dissipated and that is the purpose of the transmission oil cooler. Even in a cold climate, the transmission will warm up pretty fast. Some heavy equipment runs a thermally controlled valve to increase or decrease oil flow to the oil cooler until a steady operating temperature is reached. Unless it is twenty below outside, I would not be very worried about cold transmission oil in an automotive application, if it is that cold, my C-Body will say in the garage.
Dave