I am perfectly aware that "I" is simply not a used response to a question, in fact that's precisely my point. And not only for questions, for that matter - in a positive statement made about yourself in which you want to stress that it's about yourself, you could say "Me, I think that..." (admittedly, this is less common in English than in French where it's perfectly normal).

For whatever reason, English and French, unlike many other European languages, have determined (I don't know if English borrowed it from French, or if they came to it independently) that the nominative of the personal pronoun of the first person simply can't be used on its own; in cases where it would be used on its own, they have substituted another case (if one judges by French, it's not actually the accusative case, but the case that you get after prepositions; in English, there obviously is no such distinction). French goes further and does the same thing in all other persons (e.g. "Eux, ils disent que..." = "Them, they say that..." ), English does not and that literal translation looks very weird as a result.

That is obviously the most normal thing in the world for speakers of those languages, so it may be strange to think about it the way a native speaker of a different language would. But I think trying to find a logical explanation for it by offering all kinds of alternative answers to "Who wants a cookie?" which might somehow grammatically justify the "Me!" answer is fairly pointless. It is what it is, all I'm saying is that from a Latin-based "subject takes nominative in all cases" grammar standpoint, it's a deviation from the norm. That doesn't mean it's "wrong" or somehow needs to be defended as a weakness.