Should the radiator fan operate while the car is moving?

That depends on how good the airflow is through the radiator at the various speeds. Low speed high load is usually when the fans will come on.

Fans are controlled by sensors and ecu so that the engine temperature is kept within a specific range.

Issues like the fans not turning properly, faulty sensors can cause overheating but so can many engine faults.

Even waterpumps can be faulty and reduce the coolant flow. Radiators can be clogged due to age and/or incorrect coolant additions over time.

All has to be checked to find the real issue or issues.

Should the radiator fan operate while the car is moving?

YES. It should if the coolant is not being kept cool enough from just passive air flow alone to keep the engine temp low enough to prevent damage to the internal parts .

What I understand that when the car is on the move the air from driving should be enough to cool off the coolant inside the radiator so the radiator fan shouldn't turn on high speed?

In modern cars the electronic fan is controlled by a thermostat. (In the good ole days the fan was always being turned by the engine via a belt and pulley system connected to the crank, some fans had a clutch)

The fan turns on when the temperature of the coolant reaches a predetermined temperature. The thermostat does not know if the car is moving or if air is flowing through the radiator, it just knows that when the coolant temp reaches a predetermined temperature it then sends a signal to turn on the fan. If the air flowing through the radiator is not cooling the coolant enough to reduce the temp of the coolant, and therefore the temp of the engine, then the thermostat will send the signal.

If you are driving in Arizona and the outside temp is 110 degrees then the air is not providing much cooling as it passes through the radiator. Or if the engine is making more heat than the air passing through the radiator can cool then the coolant is not cool enough and the thermostat turns on the fan(s) to assist.