I'm confused why the declaration of an emergency might not be made to avoid paperwork. The humble incident report is part of an aviation professionals responsibility to complete whether or not an emergency was declared and its' purpose is to improve flight safety. An emergency is normally declared in order to obtain the required priority to either recover the aircraft safely or to alert rescue resources or both. If neither is required there is no point declaring it.
Anybody who cannot be bothered to complete flight safety related paperwork should not be flying military aircraft.