The term "objective" morality is not sufficiently precise. It appears more accurate to say that an objective basis for morality exists, rather than saying "objective morality" exists.
I agree that the expresion "objective morality" might not be sufficiently precise, but your definition as objective basis for morality misses the mark by a long shot.
Let me give it a try at explaining what it is we are trying to point to, here (at least me).
The statement that M: "some x is a (morally) good" , where x is some action or state of affairs that is actual, is an statement that is truth or false independent of opinion, and it is such that there is a fact of the mater that makes M true or false.
That objective basis can be found in evolution itself. Humans are not the fastest or strongest organisms on the planet -- we don't have sharp claws or teeth. One-on-one with larger, stronger predators, humans don't have much of a chance. The key to human survival, as it is with many other organisms, is the ability to work in teams or groups. A structure for social interaction is absolutely necessary for this, and would evolve simultaneously with the organism itself -- through a cause-effect process which would naturally tend to diminish the prominence of behaviors that didn't work, leaving the behaviors that did work to become more prominent in the organism.
This behavior filter is a bit messy -- because socially-cooperative behavior is not the only successful survival strategy. Aggressive or somewhat anti-social behavior also provides some reproductive advantage -- not enough to become as prominent as social behavior, but enough to remain a factor in human behavior overall.
Your account might be a runner up to explain why certain behaviour is conducive to survival or reproductivity. It does not explain why those things are good or evil.
Social behavior would naturally include "fuzzy" concepts like "trust" -- and "trust" would unavoidably include concepts like honesty / deception, theft, and even adultery...moral / ethical values that "magically" have found their way into most of the world's religions.
why would social behaviour include concepts about anything ? that still does not explain why those concepts or any other would be concepts of good things in actuality
Ants exibit social behaviour, and, nobody is lining up behind the hypothesis that they have concepts like trust, or antyhing like it.
So...there is an objective basis for morality...and it's tied to survival itself...no God required.
You have described a hypothesis of how social behaviour might be conducive to surivival, and pressuposed that such social behaviour would form concepts, which, even if true, still does not explain why those are good or bad.
Just to be clear, you need to say something like: surviving is the good , or, reproduction is the good, or social behaviour is the good, or something like that, and that statement must explain fittingly the truth of moral goodness, beyond opinions.
So far, you said that such and such behaviours work, lead to reproduction , to surivival, but why are these things good at all?
Your account pressumes, inadvertently, some of those concepts are concepts of good things, before the account even gest to the starting line.