It says Brown can buy a gun for Black as a gift and that Brown could do the paperwork because Brown is the person making the purchase. In this case, the husband has already bought and paid for the gun as a gift to his wife. The ATF rule from your post 6 says the husband could lawfully complete the paperwork because he is the purchaser. The question is, does the ATF rule that says the husband could do the paperwork also allow the wife to do the paperwork, since it is a gift to her?
From reading the information from the ATF and the form 4473, I thought the person paying for the gun always has to be the one doing the paperwork.