The fan will spin just a little when you shut off the engine. The clutch doesn’t engage until the airflow thru the radiator gets warm enough.
You can test fan function pretty easily. Throw a heavy blanket over the grille, the idea is to restrict airflow thru the radiator. Start the engine, hold the engine at a elevated speed. Watch the temp gauge, don’t overheat the engine. After a while, there should be a very noticeable increase in airflow, along with a loud roaring noise. Once you notice the fan noise/airflow, let the engine idle a d remove the blanket. After several minutes, the noise should stop, along with the airflow.