Yeah, but this is someone who's robbing the bank. They're not going to identify themselves unless they want to appear on one of those World's Dumbest Criminals shows.

The policy would have to make exceptions for bandage scenario, but not only is that a five or six sigma event, but that person is already conspicuous, and can be given special attention by the bank staff (perhaps asking for identification) before they can leave a note demanding money. Someone who just wears a hat with a brim that can cover their face is in a grey area where they're not quite conspicuous enough, but they do have a reasonably effective way to obscure their face.