I wouldn't put Woody in Chaplin, Keaton or Sturges' league, either (I have difficulty with Tati, because of his lack of narrative). Werner Herzog once said that Keaton wasn't so much a filmmaker as he was an athlete, and thus, I'd say he's the most ingenious of the list. Still, I admire that Woody moved from knockoffs in the Marx Bros/Bob Hope vein to a more complex and personal style of filmmaking. Not that I have anything against Marx Bros films, I love them, but Woody's somewhat autobiographical, soul-searching films have the mark of a real auteur (to me).
I'm sure Woody would be the first person to admit that he's a borrower, not only only from cinematic influences, but literary ones, too (most notably Chekhov), but I think there's a substance and depth to his best films, a "measure of sophistication and personal complexity" (All Movie Guide) that make him stand out. What's ironic, perhaps, is that the films he's most proud of (and there's very few, incidently) are amongst my least favourite -- notably Stardust Memories and Husbands and Wives. When you hear him talk about those films, and what he wanted to achieve, they sound great, but he doesn't seem to be able to realise those meditations on screen, and the reason for that may be as you've stated: that his visual scope is extremely limited. I would much rather watch him tell stories in Radio Days than try to be half a Fellini, and he does much better pastiches of his favourites, anyway, as is the case with the mordantly witty Love and Death.
Casting his aspirations aside, when he has a great story to tell, his camera does an excellent job of telling that story. As a director, in terms of shooting to tell the story, he's brilliant. It would appear that he's a great director of actors, at the least he's had some great working relationships with some talented people, and he's had some fantastic people crew for him (be it Gordon Willis on cinematography or Santo Loquasto on production design). To be able to write, direct and more often than not, act, is an impressive accomplishment, and has done a lot for lowbudget filmmaking, both independently and within studios. I'm happy to be amongst his fan base.
If I were to compare Woody Allen to anyone on your list it would be Sturges. While Sturges may have been more brilliant than Allen, they were both able to make thematics, character relationships and great storytelling the focal point of their comedy.
I never noticed the lack of blacks and latinos in Manhattan. I guess it's "his" Manhattan, come to think of it, I can't think of many characters in his films who aren't Jewish or Italian, so I guess he writes about what he knows best.