'Evolution' does not necessarily imply evolution by genetic natural selection. Humans evolve mostly by way of cultures of belief .I'm sorry, I don't think this argument stands up. You argue that "evolutionary reasons why various moral codes emerged" may suggest a common theme in our moral codes, and I agree. But what reason is there to believe that this commonality is universal? For "objective", in this context, carries with it "universal" as one aspect of its meaning, doesn't it? To equate commonality with objectivity is no different to asserting that consensus = objectivity, as far as I can see. And that latter is easily refuted by remembering that there was a time, not too long ago, when the consensus was that the Earth is flat, but objectivity said otherwise (even if we didn't know that at the time).
For instance Muhammad, a well -travelled trader and business man ,saw that the Arabian tribes' vendetta method of social control which was based solely upon reciprocity was disfunctional. So Muhammad introduced Allah the One God whose moral code was , besides reciprocity, based also upon universalistic intentions before the event.
Earlier monotheistic religions had been developed centuries previously within several geographically distinct civilisations where prophets had understood the value of the Golden Rule for social control. The Golden Rule is universalistic not tribal.
It's reasonable to presume universalism as a main part of morality developed as a reasoned response to economic changes towards people living in larger and settled communities, changes that happened in various localities within the time frame of about five centuries .