"Mayday, Mayday, Mayday" tells ATC (and everyone else on frequency) that you have an emergency, but it gives them exactly zero information about the nature of the emergency or what you need other than getting their attention and time on frequency. If you're in a portion of flight where you don't already have the attention of the controller, then adding "Mayday" or "Pan-Pan" calls to the beginning of your transmission helps to get you that immediate attention. However, if you already have their attention and other people aren't trying to talk on frequency, then it's not as necessary.
Furthermore, when quick communication is necessary and/or the pilot needs to be focusing on flying and navigating the aircraft rather than communicating (as is often the case in an emergency situation,) adding the "Mayday" calls to the beginning of your transmission even when you already have the controller's attention may go from merely unnecessary to actively unhelpful.
Of course, when you're in areas where English isn't the native language (or if English isn't your native language,) then sticking to the standard phraseology becomes more important, both for the sake of ATC and the other traffic on frequency. Even when all parties involved are fluent, significant differences in accents can make communication less clear, so using clearly-recognizable standard phraseology is more helpful for clear communication.
As far as the situation in the linked video, I would say that was more a case of ego than unclear communications. There's nothing particularly unclear about,
"We can't land on 22, we're breaking off the approach, and if you don't give us runway 31R, we're going to declare an emergency."
The situation wouldn't really have been made any more clear by adding "Pan-Pan, Pan-Pan, Pan-Pan," or "Mayday, Mayday, Mayday" to the beginning of it and everything they did say would still need to have been said. Once American 2 became aware of the crosswind problem on 22L, they quickly needed to change to 31R and they needed to communicate that as quickly as possible to Tower. Adding something to the beginning of the transmission would have just made it take longer to communicate and at a time when pilot workload would have been quite high preparing to change runways near the airport. The pilots needed to be spending their time talking to each other and focusing on flying and navigating the airplane, not on talking to Tower.