You are correct that a Creator/God isn't necessary to oppose a moral system contrary to yours, but moral absolutes that exist outside the human mind are necessary to ground those morals or rights in something that can be defended as objectively superior. That's where your world view falls on its face. You can oppose my decisions as king, but you can't objectively tell me I'm wrong. You would say we should have battled the Nazi's and Imperial Japanese in WW2 and that they were wrong, but your morality based in evolution foundationally and then culture, reason, and experience can't tell me why they were inherently evil even, though I agree with you. All you can say is because we should avoid human suffering. I agree, but who determines what suffering is and why I shouldn't do that. Are you the arbiter of it? Is it a group dynamic? Is it evolutionary? Is it a combination? Why should I listen to you? What makes your view better than Mao's?
You really admitted my position is the correct one within your post: "Evolution may help explain why humans developed empathy, cooperation, and concern for others. Human moral systems were then shaped by reason, culture, experience, and the consequences of our actions."
My reason, culture, experience and the consequences of my actions can lead me to a totally different version of morality, and that's fine according to your view of morality if it is taken to its logical conclusion. Evolution may have laid a groundwork, which I granted in my king analogy because that moral frameworks highest duty is to family and then to a lesser extent the wider nation/culture. It is contained within the group.
That is also seen in animals all over the planet. A group of chimpanzees will defend and take care of their own group and will savagely fighting another group. They will literally eat the young of another group. If an alpha chimp needs to defend his position he will fight and kill if necessary be that outsiders and even those within the group, and likewise others want his position and will try to kill or drive him away. We see the same thing in lions. The internal group is primary and even then it can and will go a different way within the animal kingdom. We are just more evolved primates in your view so the same is fine for us as well from a subjective moral viewpoint.
That's where your view leaves you. It leaves you as an individual with overall good morals and values that you can't ground in anything outside of yourself. It comes down to because I said so or because the group said so. Well other groups like radical Islamists have a different because I said so and you don't have a leg to stand on from an objective moral viewpoint. You have no choice but to admit that your view isn't the only valid one and that all views have equal validity even if you don't personally agree with them, which I know you don't, because the underlying evolutionarily development and subsequent cultural and experiential development, which is your position, can lead another individual or group to come to a different conclusion, and you can't tell me why it's wrong.
You were born on third base morally, you hit no triple. The basis for your entire moral worldview is a rip off of mine with no foundational underpinnings that exist outside of your own mind and personal experience. It's a house built on sand.