Being "correct" is not always being sensible
That's a fair point. Sometimes we lose sight of the common sense in our eagerness to do things by the book.
Case study number one: I was controlling the Scottish Sothwest sector a couple of weeks ago when Shanwick phoned to say that the BA LAX-LHR flight was probably going to declare a medical emergency upon reaching NIBOG (the boundary between oceanic and domestic airspce). Pilot of said aircraft called up about eighty miles out and in our subsequent discussion he stated his intention to declare a medical emergency due to the condition of the passenger (which I won't go into on this forum...) though he would be continuing to LHR. At NIBOG I asked him to squark 7700 and arranged a direct routing from there to BNN. This took him through Shannons airspace (just a small bit as it happens) and coordination was affected with the Irish. Aircraft was handed off to LACC and as far as I was concerned, that was it. Hopefully, the passenger was OK.
Case study number two: The very next day a VS flight from LAX-LHR declared a pan due to a medical problem on board The controller (it wasn't me) squarked him 7700 and arranged a direct routing etc.
Both cases were handled identically despite one being a 'medical emergency' and the other being a 'pan'. Whether either case warranted a full emergency I've no idea - the fact that both flights continued to LHR would maybe suggest not - but that is not our call. If you feel you have a problem, we'll help, no matter what is declared.