I'm not sure why people are fixating on the issue of yelling Mayday. ATC can only do so much and they certainly can't fly the plane for you. While the communications seemed chaotic and messy to those listening afterwards with full knowledge of what was going to happen, it wasn't that abnormal. Comms get stepped on all the time and although the radio check request was a bit unusual, I see no indication that ATC didn't do everything they asked for. ATC recognized the emergency as such very early on, as soon as he said "lost an engine", offered the immediate return and then just let them work their problem. All this talk about proper terminology misses the fact that the term 'emergency' covers a wide range of possibilities and if you are already in contact with ATC--as opposed to breaking in out of the blue--then skipping the formalities and letting them know the exact nature of the emergency seems more helpful. If the controller hears "bxxxt hmmmm blllll ENGINE grrr blllrtt IMMEDIATE RETURN hsssmmm", they'll probably get the picture and start clearing the way. If you truly can't get through to them after 1 or 2 attempts and you need their immediate attention for some reason--like needing to turn in a busy terminal area--then squawking 7700 and yelling Mayday three times may be appropriate. I just don't see how any of that would have helped here. Yes there was additional confusion with the vectors for the other aircraft, but I still see no issue. The only minor possibility is that the comms mess distracted from other tasks, but that's the pilots job to manage.
We have nowhere near enough information to criticize anyone's performance here. For me the biggest questions are why did the second engine fail and at what point did the crew realize (or should have realized) that either the engine was malfunctioning or that they could not hold altitude? Perhaps the first engine failure occurred at rotation and they melted the second one trying to climb out, or perhaps that second engine wasn't in good shape to begin with. Unfortunately all that evidence is now at the bottom of the sea. I hope they make the effort to recover enough parts to do a thorough inquiry.