Ben_campbell, I do agree with almost all of what you wrote. If you had asked me a couple of months ago about Shirley Bassey, I would have said she's the singer of Goldfinger and Diamonds Are Forever, that I'm a James Bond fan and I really love those two songs, but that's all I know. I wouldn't have known how to classify her at all. When I think of "old school" popular singers I think of Tony Bennett, Johnny Mathis, Andy Williams, Perry Como, Peggy Lee, that is, as you say, music your grandparents or parents listened to. But I didn't even know enough about Shirley to place her at all.
But now that I have been introduced to her live act through the YouTube clips, I have just gone nuts over her, and I do think that she very much knows how to inhabit a song, take for example "I (Who have Nothing)." (And who can compare with Sinatra--he was in a class by himself, perhaps the greatest popular singer ever!)
But despite "Get The Party Started, "The Living Tree," and playing at Glastonbury, she does not enjoy the same current popularity of, let's say, Neil Diamond, who played Galstonbury the year after Shirley. Neil can still sell out Madison Square Garden two or three times over.
Trying to explain why I have become such a big fan so quickly is like trying to explain why I became a fan of the Grateful Dead after hearing the very first note, and how when I went to see them 30 years ago, the moment they started to play, despite my huge expectations, a big smile crossed my face and I thought, THIS IS IT. This is my kind of music. I've had the same experience with her--and as I don't expect everyone to "get" The Grateful Dead, I don't expect everyone to "get" what I see in her.