With a running hot engine, if you pop the hood, the fan should be pulling plenty of hot air thru the radiator into the engine bay. And if you tried to stop it (the engine fan), with a roll of paper towels or something else that's not your hand and unlikely to break the fan blades, you should encounter plenty of resistance and it should just continue turning. If the airflow is kind of weak, and you can actually stop the engine fan, you most likely need a new fan clutch.
Or, with the engine off, you can compare how hard it is to turn over the engine fan by hand first thing in the morning (cold engine), and compare that with how much harder it is to turn when everything is hot. It should turn relatively easily when cold, and you should feel significantly more resistance when hot.
I've had this problem on other cars that only have an electric radiator fan. When the fan died, there A/C was definitely not cooling very well when stopped, but fine while out on the highway. And no, the coolant temperature gauge wasn't moving from the middle. Many of today's cars artificially keep the temperature needle right in the middle even as the coolant temp. varies. and only shows cold or hot when it is way cold or way hot.