My problem isn't the diet drink (I just ask for tap water if there isn't any that I like), it's the coke itself. Any sort of coke.
I used to work in a lab that tested food additives and you wouldn't believe what goes into both coke and pepsi. Have you never cleaned a bath with coke, that should give you an idea of what goes into it.
My daughters weren't allowed either until they were old enough to decide for themselves. One still won't drink them and doesn't allow her own children to either, the other does occasionally but did the same as me with her kids.