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. The most controversial part of your viewpoint is that morality has no exaptations or emergent functions, that its essence is social coordination and that all the rest is hot air.

So that's why the question of whether your game theory reduces in turn to instrumentalism is important. Because it would be dubiously reductive to suggest that instrumentalism explains virtually all human behaviour. Instrumentalism is neutral on the question of which ultimate goals or desires we should pursue.

A computer programmed to be perfectly instrumental would happily guide a serial killer in choosing the most efficient means of carrying out his evil objectives. But there may be something evil about murder that doesn't show up in the amoral instrumentalist's analysis.

If you're being instrumental about morality when you explain that behaviour in game theoretic terms, that's fine as far as it goes. But instrumentalism opens up other questions that call for higher-level explanations, such as the question of why we're driven to achieve goals in the first place.

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? Or why do we prefer to live in peace? Perhaps doing so makes it easier for us as individuals to achieve our private goals. And why do we care about fulfilling our private interests? Why are they worth pursuing?

When you ask enough questions, I suspect you'll find that the instrumental analysis bottoms out in ultimate values, not in means-end calculations. One of those fundamental values is about something like the sacredness of life. If you think there's nothing interesting philosophically to say about our choice of ultimate values, I'd treat that as a scientistic prejudice.