To “do unto others as you would have them do unto you” implies cognition to me. It requires an organism to think about the way it would like to be treated. I don’t think it can be argued that other animals are capable of doing this. Most animals satisfy their basic drives - or just respond mechanically to stimuli. Examples of charity and empathy in animals are generally explainable by evolution, but they usually are not equivalent to the golden rule (“charitable” animals will save their own offspring, because that propagates their own DNA, but saving another’s offspring would not, so that type of behavior does not occur in extant species).
In any case, I am an atheist and don’t believe in a “morality” (I don’t think that the golden rule would rigorously enough prescribe what course of action I should take in a given circumstance) The golden rule will lead to conflict in the event that I want to be treated differently than someone else. Consider abortion. According to the golden rule, any atheist who wants an abortion would have to be willing to have been aborted. Maybe some would concede that point, but I think many would not.
Atheist “morality” seems, rather, to me to involve cooperation according to mutually agreed upon subjective valuations. I don’t think that secular society is very willing to phrase it in this way, but I don’t believe that atheist morality could be “more” than that. Consider abortion again. Atheists are willing to get abortions (to speak very generally) because they have decided that killing a fetus is an acceptable course of action when not doing so would represent an undesired burden.