It would have sounded different, for sure, but not necessarily "better'.... Jenkins' "sentimentality" was a known quantity, and apparently a desired one as well. When it goes over the top (my "favorite" example is Cole's "Love Letters", which made me queasy - literally - the first time I heard it), it's pretty hard to take, but when reigned in/disciplined/applied judiciously/whatever, it's a genuine "flavor" that is a legitimate as anything else. I think it works just fine for the material on September Of My Years. That's not exactly an unsentimental album, and Riddle, great as he was (by far and away my favorite of Sinatra's "Big 3") would probably not have nailed it like Jenkins did.
I think it might be fair to say that the POV of Jenkins was one that was decidedly non-urban, non-"hip", and non-"ethnic". In other words, the guy was pretty damn WASP-y. But that's not a sin in and of itself, I wouldn't think. I do think, though, that it presents a problem, or more accurately, a point of reference, which many people, especially today, find disfavor with just because it is what it is. Too bad, because that whole vibe is just as real, and just as nuanced of a mixed bag, as is anything else.