Does IOS "Find My" App work when the app is not open? If so, why?
Sharing your location with other people and locating a lost device always works, even when the app is not open, providing other toggles are enabled.

The ‘Find My’ app location services access is separate to the system service that shares your location to other people in the background.
The toggle for the background sharing of your location to friends you have shared your location with is near the top of Location Services settings, called ‘Share My Location’.
The toggle for the background sharing of your location when looking for your own lost device is at the bottom of Location Services, under System Services, labelled ‘Find My iPhone’.