If you check on sites like http://avherald.com/ you will see how many incidents are worldwide - most of them are only incidents (and ending in a safe landing), but even then they will be considered an emergency while still in the air.
Isn't the purpose of an emergency call to be used only in really critical scenarios? How is an aircraft going down to be differentiated from one carrying a sick passenger?
The purpose of an emergency call is any situation where the aircraft needs special considerations for it or its occupants to remain safe.
If the plane is going down, the pilots will be ...