Extreme anything, heat, cold, altitude, will show any weakness in a cooling system. Bad thermostats, faulty fan clutches, dirty radiator fins, clogged radiators, cracked radiator cap seals, all will show up when in extreme conditions.
A fan clutch is worn or seized bearings can put the fan into a permanently "spinning too slow" mode, which will mean the engine runs too hot at idle, but airflow whole moving at higher speeds is enough to cool it down.
A radiator cap with cracked or worn seals will cause a slight pressure drop inside the cooling system, which raises the coolant temperature. It's most evident in high heat or high altitude.
At operating temperatures, if one side of the radiator is too hot to touch while the other side ain't that bad, you've got a clogged radiator that won't cool by radiator fan alone, and needs high speed winds to assist.
Warm air conditioning may be a symptom of a failing AC compressor or something else in the AC system, or of the cooling system. There are millions of vehicles that operate in high heat and their air conditioning works just fine without having to add a second, separate electric cooling fan. Adding a second fan is treating a symptom rather than the cause. I'd have it checked out.