There's no competition. I just want to make sure the facts are clear.
I'm sure that any Japanese artist who was specifically a jazz
singer, as Martha Miyake was, would emulate the sound of American jazz singers. Japanese adoption of aspects of American culture has been commonplace for a long time. And it did sound to me like some of the pronunciations were either a bit off or a bit too precise, as if the song were being sung by someone who was not a native speaker of English.
Some of the dubbing was unconvincing, yes, but that's just normal lip-sync problems. A lot of people aren't aware that virtually every time someone sings onscreen, the voice track was recorded beforehand in a recording studio (for optimal sound quality), then played back on the set and lip-synched by the performer, even if it's the same person who actually sang the song. Some people are better at doing the lip-synching than others. So yes, it is a dubbing problem, but not the same type of dubbing as when Japanese dialogue is replaced with English dialogue. It's understandable that the two would be confused, but if you read her lips, she's clearly mouthing the English words, even if she sometimes isn't mouthing them exactly in synch with Miyake-san's voice.