So what are you supposed to say instead of "can I get"? "Can I have"? Or something else? I had no idea this was considered "American".

I suspect that most of the "hate" comes from a small group of sour-faced vinegar tits who just love to put their pet peeves front and centre every chance they get to "correct" people, so they can masturbate about how smart they are and how dumb the unwashed masses are. Meanwhile, most people could care less.