Medical emergencies far exceed those of a "mechanical" nature. Rarely a day goes by at Heathrow without one or more.
I think the thread also brings to question the need for any sort of magic code word nowadays. Given that current R/T comms are better than a telephone, a plain language message from the crew should suffice. If someone says to me "We have a sick passenger with a possible heart attack" I don't really need him to do the "PAN, PAN, PAN" bit! Equally, if he says "three of my four engines have flamed out" I don't go back and say "If you are declaring an emergency will you kindly prefix your call with 'Mayday'" Maybe in centuries to come it will all change?