But you wouldn't describe the death in office of a president as "The End of the Presidency," generally. It may not be wrong but it makes them sound a bit hysterical.
But using the definition that "the papacy" describes a particular reign or administration of a certain pope, the end of it is always by death. Popes don't leave office the same way as presidents.
ETA: I mean, I don't see anything wrong with a headline at the end of an administration saying, "The End of a Presidency."