If somebody is mumbling, it's difficult to understand the content of what they are saying. Mumbling is down to poor diction.It's not unknown for mumblers to be asked to speak more clearly!

If somebody is singing, the combination of vowel sounds, and pitched notes make it even more difficult to understand words (listen to an operatic soprano - normally the vowel sounds are mutated in order to make the top notes easier to sing ... words are quite often indistinct ... and infuriating).Therefore clear diction is a necessity if the listener is going to be able to understand what words you are actually singing.

The major, if not whole, point of singing a song is to tell a story. To do that effectively, your words have to be heard. So yes, good diction is an essential. I'd repeat Brian's final sentence at this juncture "If you're singing,say, a traditional ballad, it loses its point and the audience's attention if you fail to get the words across." I have known occasions when I have "switched off" and wondered what the point of being present was if the performer couldn't have the courtesy to address a little care and attention to trying to communicate to his/her audience.

As to whether one needs to know who the OP was referring to ... probably not (although I can think of at least one candidate for the vacancy),as the principle is a constant.