I think there is nothing else that can be said at the beginning of a radio transmission other than "MAYDAY, MAYDAY, MAYDAY", which will immediately silence everyone and make them pay extreme attention to what's being said next. I think this is the most important strength of using that phrase. Not only ATC but also everyone else will immediately understand what's going on.
Real life communications of US emergencies go like: "Uhh center this is American 123, we're out of Dulles through 6 climbing on to 10 and we need to stop our climb at this point, we are declaring an emergency due to smoke in the cabin and... uh... we'll request to return to Dulles after we run some checks first, can we take up the hold here over the VOR at 10?", in a very calm and standard way. Anyone else hearing this in the background won't pick up that they just declared an emergency because it's embedded within the transmission as another piece of "normal" information. It seems routine.