I see no reason why morality wouldn't perform that function of social coordination. But I think it would be hard to prove that that's all morality does.

No harder than proving any negative, which is to say meaninglessly impossible. Whereas a simple counterexample would prove the opposite. You seem to think you have offered some, but I have to admit that I don't understand them.

Instrumentalism is neutral on the question of which ultimate goals or desires we should pursue.

As is game theory. That's why moralities vary from environment to environment. Scarcity is the main determinant, because scarcity determines the sum of material well-being available to be created and allocated. Where material prosperity is not available, virtue literally becomes its own reward.

I have offered you a difference between instrumentalism and game theory, which you have chosen to ignore. Game theory explains the survival of accidentally selected strategies, like mutations and superstitions that become customs that become mores. No one is trying to achieve anything, but if those following any other strategy perish, only those following the "fittest" strategy survive, perhaps unaware that any particular strategy is the reason they are still around.

You want to say we prohibit murder because murder is inefficient in helping us achieve the goal of living together in large societies. Talk of the so-called sacredness of a person is just a rationalization of that instrumental calculation. Why, then, do we want to live in large societies?

It's economically efficient, it maximizes choices of mates, there is safety in numbers, or maybe something else. But Hobbes's first downside to the warre of all against all is that it makes life "solitary." Maybe that's enough to make life poor, too. It's a strategy and it works, so well that we are genetically programmed for it. No one had to dope it out as an instrumental strategy. We may well have inherited the trait from chimps,

When you ask enough questions, I suspect you'll find that the instrumental analysis bottoms out in ultimate values, not in means-end calculations.

That's right. But then, instrumentalism is your idea, not mine. I'm explaining why strategies survive, not why how or why they come into existence. The "values" are post hoc reifications. Put simply, I have no problem with Euthyphro's paradox. God loves the pious because it is good. Indeed, God only "exists" because the good needs someone to love it.