Well, I've written the first draft of an article on game theory's model of morality.
Your distinction between game theory and moral philosophy is certainly charitable to philosophy, but I suspect that game theory's approach casts all the moral theories in a liberal light, as it were, since answers to moral questions become so many searches for social conventions and compromises. One way or another, if game theory says anything worth saying, it deflates morality.
Your take on what game theory says about morality, as distinct from what philosophy says, comes perilously close to making game theory's model trivial. Maybe this is like arithmetic's take on a dozen eggs: arithmetic assigns numbers to them and says how many eggs you have left if you remove some or add some others. As for telling us what an egg is, you'd have to look elsewhere.
Still, because the medium is sometimes the message, it's possible, as I said, for a superficially trivial discourse to have non-trivial social effects, by narrowing the Overton window or by having certain connotations, given the proffered vocabulary, which mold behaviour to suit certain institutions. Calling morality a social convention may seem innocuous, but it's not since it's deflationary.
[Sorry, I'm posting this again because my reply got accidentally pinned to my Medium homepage.]