I suspect that calling it as an emergency rather than a mayday situation reflects the fact that they initially regarded a single engine failure as an annoyance to throw a checklist at - which, to be fair training and ETOPS thinking confirms.
At such low altitude though, scrutiny of the second engine and fuel situation would be the first 'survival' priority - Once they had the first clue that they weren't able to continue the climb as expected for OEI they should have immediately focussed on the fastest return vs nearest landing.
And yes Mayday might also have cleared the frequency a bit
Last edited by Jetstream67; 4th Jul 2021 at 08:38.
Reason: typo