Bennett's connection to jazz, such as it is, goes back a long way. In his teen-age years in Queens he sang at sessions with Al Cohn, who remained a musical associate over the long haul, and on Bennett's very first album, "Cloud 7," from 1954 he was backed by a small jazz group led by his friend Chuck Wayne, with Dave Schildkraut on alto. That's not to say that Bennett is a jazz singer; he's not. Rather, his singing is compatible with jazz accompaniment, and he's usually stimulated by it. The best example of this IMO are the two albums he did of Rodgers and Hart songs with Ruby Braff for his own label Improv, later collected on Concord.
I would also add The Beat of My Heart to that list. Even if you disagree that the execution wasn't perfect, you have to admit that took Jupiter-sized cajones to be a pop singer and make a very decidedly non-pop record back then.
Completely agree with the statement in bold as well.