"Mayday" is the internationally agreed code, so that's one reason to use that - but US accident investigators also noticed a pattern that American pilots are often too reluctant to declare emergency status when it's appropriate as it is here, because they feel it's "not that bad" yet. Having them say it's an emergency without saying "Mayday" is good enough and might mean a few more cases that should have been emergencies are declared as such.
I might look for the link later, but there's an incident with a Youtube reconstruction where a private pilot is nearly out of fuel, the weather is poor, and he's approaching a USAF base en route to his last hope landing site. Airbases are military facilities, closed to civilian traffic. He asks the controller if he can land there. She says he can't... unless it's an emergency.
He should say "Yes, this is an emergency, I'm on fumes here and the weather is much worse than I expected". She'd turn on the airfield's powerful landing lights, and maybe he'd spend the evening explaining his screw up and apologising to a base commander or at worst spend the night in a cell. Nobody dies. Instead he pressed on, and his dire situation only became clear to her when it was too late and he was already doomed.