Actually fans come MUCH ABOVE radiator. And in fan there are two fans in your car: engine (clutch) fan and AC fan.
AC fan works on electric as soon as you switch AC on it supposed to start to cool engine from extra load of AC.
Engine fan works with belt, that is running on slow speed due to viscose clutch in the fan. When car overheat little bit this fan supposed to work on high speed and bring the temperature down. For checking this when u pick your car in morning and when its cold for first 5 or 10 seconds engine fan shud run on high speed and make big noise as a self check, same as all lights in dash appear for 5 secs on first start and then dissapear. If its doing that, that mans fan is fine and you need to look somewhere else.
Ask your mechanic to change the clutch of fan and not whole fan it will save a lot. As viscose clutch is the main culprit here which gets failed due to age, nothing else.