You mention the "right" thing numerous times, as if there is an objectively "right" thing to do in all circumstances... Given so, what is the grounding (source) of where that "rightness" originates from? How did it "come into existence"? If it doesn't transcend humanity, then how can it be objectively "right" for all of humanity?
Also, "innate morality" seems like what Christians claim when they claim that morality is "written on the hearts of all mankind".
It's "written on the hearts of mankind" the same way that attraction to the opposite sex or attraction to certain foods is "written on the hearts of all mankind"- social behaviors and emotions like empathy, cooperation, sacrifice, loyalty, etc... are hardwired into human brains through evolution. And it's not just humans. Many animals, from ants to elephants, also have it. It turns out that animals who don't have such traits don't do as well in the evolutionary fight for survival as those who do. The idea that nature is just a simple survival of the fittest and always "red in tooth and claw" is too much of a simplification. Sure that's also an important part of nature and survival. But nature weaves a far richer and more complex and sohpisticated tapestry than that.
Scientists have even localized such social emotions to certain parts of the brain. These brain centers are localized to parts of the brain involved in emotions- from disgust when you see others being hurt or injustice done to them, to pleasurable emotions when you help those who need it or relieve their suffering. They even know the particular neurons involved in such brain centers, called "mirror neurons", which are how your brain can look at the pain, suffering, or sadness in someone's face and mirror those emotions to your own brain. Have you ever felt a deep and visceral feeling of revulsion and feel like throwing up if you see someone getting tortured or hurt? Injustice being done? Yep, that's those mirror neurons at work. They can be very powerful.
We even know what happens in pathologic states where those brain centers are absent or dysfunctional. It's not too different than a dyslexic having a tough time reading. These folks present clinically as psychopaths. There have been imaging of these studies comparing their brains to normal control groups when shown pictures of people being hurt. The normal centers of disgust and revulsion don't light up. In fact in many of them, ominously enough, their pleasure centers light up. They like it. Scary stuff. But that's what happens in disease states where the brain is clearly effected. Reading scripture to such folks has not been found to be very helpful.
Humans have an innate normal capacity for love, empathy, loyalty, etc... it's as much a part of their biology as their drive for food, water, and sex. Like many innate traits, upbringing and training can also be hugely helpful. But without the biological substrate, there is as much hope to teach them anything as there is to getting a man with serious eye disease to see or a dyslexic to read.
There are cultural differences too, of course. For example, when Europeans first encountered natives in the Polynesians, the Amazon, or other places who did not cover their loins or breasts, they thought they were "immoral" and "indecent". The natives were conversely puzzled by why these Europeans liked to wear so much junk in the heat. The Muslims, on the other hand, could never figure out why the European women liked to walk around without covering their face and hair. That was even more immoral than not covering their breasts, wasn't it?
But these are all just cultural differences. Some practices are more dysfunctional than others, and it has a lot to do with things like the weather and the contingencies of culture. These things are very subjective and can change across geography and time, and don't matter that much. But the root of all morality is about empathy: the ability to see and feel someone else's pain, and be bothered enough by it to do something about it.
Divine command theories of morality fail, because it just becomes a giant game of "Simon says". "Don't torture young children" would be no different than "tap your head and rub your belly at the same time". They are emotionally and rationally meaningless. It requires you to shut down your own natural emotions of empathy and your own judgment and intelligence, and become a robot which just follows random commands. That's not where our morality comes from.