I had something very similar issue few years back and it turned out to be the Engine fan was the culprit.
Engine fan works with viscose clutch and that usually rotate at half rpm speed of engine but when engine gets hot then viscose clutch get fully engage and rotate at 100% speed of engine RPM. This help drawing bigger amount of air and pushes through radiator to cool the coolant at a faster rate. The amount of air and pressure it uses is far above any electric fan and it is capable to dissipate heat within seconds.
Engine fan usually have a life of 5-10 years depending on usage, and it's slightly expensive to replace somewhere in range of 800-1200 AED.
Go again and test your car in desert and pay extra attention this time when car overheat, that engine fan started working at higher rpm? As it's mechanical operation, so there is no proven and clear way to see when it works except you need to listen that result in LOUD GUSHING NOISE from engine bay. If your car still be quite when engine is slightly overheating then its confirmed that this engine fan viscose clutch needs replacement. If this fan produced enormous gushing noise and temperature drops and then this noise goes back to normal noise level, then that means this engine fan is working and there is something else to doubt.