OP, how many of the 100 deliberate errors in the Williams’ link did you spot casually? Is this paper well known in linguistic circles? It is none too kind about Strunk, Fowler and Orwell who have been attacked at wordorigins often.
But even if that were the case, it makes sense for him to ask whether it is possible for him to obtain it. He’s not asking permission: he wants the person to bring him something.
I suppose the most direct thing would be to say “Bring me a” or “Please bring me a”.
There’s the notion that can refers only to ability; one should use may for permission. Not that I believe it, but it is there.
May I have and can I have are different registers; the latter being more informal. There’s nothing “wrong” about saying can I have; it’s just not quite as polite (but far from impolite).
In my youth adults would get on our cases for using can when they thought we should use may. One of the little things that I believe foreshadowed my interest in words and language. I noticed that they themselves generally used can in the very context they were at us about. The only time I heard it used the way they thought it should be used was in the game we called “Mother may i?”.
When a teacher at my benighted school was asked “Can I go to the the toilet, sir?” he would reply “I’m sure you can. The question is whether you may.” How we laughed. I used to have to ask my parents “Please may I leave the table?” - received manners. What do kids now say? “I’m done here. Splitsville?”