Aha! How about this. Standard usage "We don't do that any more", with "any" meaning it's governed by a negation.

But that can be reinterpreted as "We don't do that these days". Therefore, in these varieties: "any more" = "these days".

The one-or-more word thing could come out of stress. "Are there any MOre" is a different pattern from "We don't do it any-more".