Andrea Yates would have drowned her kids regardless of religion.
Do you have a cite for this? It's possible--I don't know enough about the specifics of her illness--but I think that religion is likely to have greased the skids.
[1] What if, instead of thinking that "God" wanted her to kill her children, Andrea believed that the voices in her head were aliens beaming their commands to her with an orbital mind-control laser? She might have killed her children anyway. But: there is a considerable body of literature and movies centered on the idea that humans can, and
should fight evil extraterrestrial invaders. Not only is standing up to enemy aliens
right, we can also
win, or so our society's stories on the subject assure us. There is a chance, perhaps a small one, but still a chance, that she might have gone to somebody and said "Help me! Aliens are trying to make me kill my children!"
However, since it was the voice of "
God" telling her to kill her children, the situation was different. It is a pervasive and socially-acceptable idea that "God" is: 1) invincible, and 2) owns the patent and trademark on all morality, so that whatever "He" says goes, because it's "Him" saying it. Humans aren't
supposed to fight "God," and any attempt to do so is inherently futile. It's all packaged together as part of the big-G "God" concept, which is what makes this particular religious idea so dangerous. And furthermore, killing children at "His" command is explicitly sanctioned, even regarded as heroic and noble, wherever the Abrahamic monotheistic religions are predominant. The "Abrahamic" religions are those which trace their foundations to Abraham, who became the revered "father of faith" precisely because he was willing to kill his favorite son without hesitation when a voice in his head told him to.
Since it's the voice of "God," rather than space aliens, fairies, demons, etc., it
automatically receives a moral blank check from just about everyone in our society. Most people would hem and haw and try to wriggle out of saying that they would kill their own child at the behest of a disembodied voice, but few would condemn Abraham for his deed. Likewise, they would also try to make excuses for Jephthah, who, since the child he was called upon to kill was a mere girl, did not receive a "Haha, just kidding" happy ending like Abraham did. In addition, "God" is the only entity that has eternal torment in his arsenal. If drowning her children
really was the only way to spare them from everlasting torture, then it arguably would have been a necessary act even if one does reject the moral authority of "God."
If Andrea had been an atheist, her delusion might have framed itself as aliens, the CIA, or some other force (or if it had come as "God" she would have had some degree of skepticism of its reality), and she might still have felt a powerful urge to obey. But, she would also have retained some psychological defense and ability to resist. There is only one entity in our civilization's entire conceptual vocabulary that is
supposed to receive automatic, unquestioning obedience and against whom all conceivable resistance is futile: "God." Had she received such malevolent instructions from anything else, the moral structures of society would have urged her to resist or seek help. It is only in the case of "God" that ordinary people will argue with a straight face, that murdering one's children on command would actually be the
right thing to do.
So, in the absence of religion, I think Andrea Yates would have been less likely to have murdered her children, and more likely to have sought help.
