Can't remember how I took my cooler out after a wreck to fix it. It was twisted like 30 degrees out of flat axially and bent in the middle as well. Careful force forming it on dead flat concrete and slowly I bent it back into the proper shape and was amazed to see it still passed a quick pressure test. Put back on car and running fine since '07. The early ones at least are separate cooler only and do not involve the rad at all.
Ford changes that cooler loop up on almost every car they make. Some cooler alone and some radiator only and then some cooler and goes through radiator too to bring fluid up to temperature quickly. I prefer cooler alone but no harsh winters here and can get away with it. I let the car get older but once I start fearing the rad popping the intercooler inside I then usually drop the radiator out of the circuit. It is stupid to lose an ATX over a failed intercooler.
FYI, if the trans fluid has gone black with debris in it the cooler may be bad, Ford absolutely says the coolers cannot be properly cleaned inside due to the internal tube finning arrangement. Well, it seems I've broken that rule too. You can use high pressure air to blow them out but absolutely do NOT pressurize it to over 30 psi like sticking finger over end of outlet to say 'burst pressure loosen' trash inside cooler.
Fans/shroud needs to be unsnapped, unplugged, and dropped down to remove from bottom. Needed for maneuvering room. Top of condenser can be plastic wiretie wrapped at top to rad support structure to hold condenser up when that rusty cross support piece is removed and condenser is unsnapped from the bottom of radiator, radiator will drop down too. Need to provide for that. Them being loose will allow you to move them at bottom forward/rearward to get space to get to cooler. Like opening your little brother's sandwich to steal a piece of ham from it.