First of all, something like “I’ll have anything” is a valid and reasonable statement that is not negative, for example when somebody asks you what you want to drink.

But further, “anything”, “anymore” and “at all” are all very different - from what I understand, “anymore” doesn’t even exist as a word in British English, and I’d point out an example of “do you have any more?” as another non-negative. I think generally “anything” makes more sense by itself than “anymore”/“any more”, and “at all” similarly needs context. But just to provide a not-really-negative example, “Do you like it at all?”, where a positive response (“yes”/“I do”) does mean liking something.

In the end, I think your arguments might be stemming from trying to apply the term to too many things, from my understanding double negatives are very simple in that they need to have two negatives. A word being general, and used mostly in negative statements doesn’t mean it’s a negative, and that the actual negative part of the sentence is redundant.