> Much of our morality and ethics are innate and stem from our nature and evolution as a social species.
Where is your evidence for this? It would seem more evident to me that the source I am aware of through my education in history and life experience for teaching of morals and ethics has been religion.
Although many here may not be old enough to realize it, countries that are mostly secular not long ago historically speaking had much moral and ethical teaching from their religions.
Attributing this to monkeys and innate behavior may account for part, but from what I understand, you could just as easily bring up monkeys in a ruthless environment where it is every monkey for him/herself. Having an established organization or organizations that teach morals and ethics is important.
Like your ruthless monkeys, someone could point to countless proclaimed adherents in jail or advocating bloodthirsty war or abusing children or embezzling community funds.
Are there any important religious teachings that wouldn't be taught in a secular ethics class?
Yes.
Much of our morality and ethics are innate and stem from our nature and evolution as a social species. Religion just took credit for it.
Even social groups of monkeys and rodents tend to avoid killing eachother and do what is in the best interests of the group.