Atheists as a group don’t have a coherent axiological system because the only thing atheists as a group share is that they don’t believe in gods.
Asking atheists to have a coherent axiological system that is assented to by all atheists is like asking Hindus, Muslims, Christians, Sikhs, Jews, Buddhists, Taoists, Satanists, Zoroastrians, Mithraists, and Isis worshippers to all share a coherent axiological system.
Why should we expect all those people agree on the same moral axioms?
Especially since no human being wakes up one morning, decides upon a set of moral axioms, and then goes about working out the consequences of these axioms and acting upon them. In real life people work the reverse way–they decide on the sort of things they want to have happen, and then try to work out moral axioms that will logically lead to those consequences.
Like, if we can just pull moral axioms out of our ass, like “Murder is wrong”, how do we judge whether that moral axiom is worth assenting to or not? In real life we decide that murder is wrong because we look at all the bad things that result from murder–revenge, breakdown of social order, our buddies dying, us dying, and so on–and decide that we better have a rule against murder.
And this simply derives from our nature as social animals. We don’t want to die because we’re animals who evolved to not want to die. We also need to protect our friends and family because we’re social creatures, and we evolved to love our friends and family. Or rather, “love” is the name we give the our evolved instinct to protect and care for our friends and family. And so if someone tries to kill us, or tries to kill our buddies, we don’t like it and we fight back.
But it turns out that the guy trying to kill us also doesn’t want to be killed, and also doesn’t want to have his buddies killed. So if we kill him, his buddies will get mad and try to kill us. And if he or his buddies kill one of us, we’ll get mad and try to kill him and his buddies.
And this leads to all sorts of undesirable consequences, blood feuds of this type are a common feature of human history and go on today in places where government authority is weak. And so the first rules of organized states and tribes is to put a stop to this sort of nonsense by forbidding people from killing each other, and work together. Of course often the way these people work together is to band together and go off and kill a third group of people who isn’t part of the circle of authority.
And since we have a rule against murder, we have to invent a reason for the rule. “Because it fucks things up” apparently isn’t clear enough for some people, so we have to invent reasons like “it makes invisible spirits angry and if the spirits are angry they punish you”. Or “the people in authority will get angry and will try to punish you, so don’t let them catch you murdering people”. Those are a lot easier for people to understand.