Eh, PANPAN is fine, but AFAIK the FAA and it's international counterparts views the use of MAYDAY in even a single engine failure as entirely appropriate.
Partly because the pilot cannot be certain that a single engine failure hasn't caused secondary damage, or that it's a not symptom of another deeper problem (i.e. running out of fuel). And in part because the FAA doesn't want pilots to hesitate calling MAYDAY and potentially making the situation worse by staying in the air longer then they absolutely have to.
The captain should have used the phrase ‘pan-pan’, which indicates an urgent situation but no immediate danger to life or aircraft.
Also squawk 7700 as said in the article, which highlights the aircraft on modern radar systems.