Surely an advantage to a Mayday is that it helps pilots communicate the nature of an urgent and unusual situation, in a pattern which they have practiced in the sim.
Both of the example video's get the message over eventually, but it takes some back and forth. No harm done, but why not learn from it?
Compared to the Thompson video (which was always going to be posted) where the pilot does his mayday and tells atc what he is doing. ATC and everyone else listens and reacts. The Pilot knows what they are going to say because they practice the start of it in every sim. ATC no what to expect, because they will also practice it. Everyone on air is clear.
And finally one of my favourites where Aer Lingus declare a mayday, and ATC ask if they are declaring an emergency...