I'm inclined to agree with Keltest (I seem to be saying this a lot at the moment). A burger may technically be a sandwich and if asked to classify it in terms of food taxonomy I would probably group it among the sandwiches. But given that food taxonomy is not really a thing that exists, in a practical context I can't imagine a situation arising where this would come up. When sandwiches are mentioned I do not picture a burger.
To a lesser extent, the same would go for bagels and filled baguettes, although I think each of those would be closer to what I think of as a sandwich than burgers are. Paninis, filled rolls and ciabatta are borderline. Croissants I think are a bit of a stretch, as mentally I group them with pastries rather than breads.
I grew up calling toasties "toasted sandwiches" and so while I would not consider an unspecified sandwich to be toasted by default, I don't think toasting it causes it to lose its sandwich status.
As to why one might make the distinction? Well, if you expand the everyday definition of sandwich to include all of the above, what do you call a normal sandwich? (i.e. two slices of bread with filling between them). You have to come up with a new term and then you've really gone off the deep end.
Standard Yankee hyperbole . Not only is the premise questionable, but conquest doesn't always result in linguistic change, and in the mooted situation, almost certainly would not have.