I think one has to ask oneself what it would be for 'morality to truly exist'. Most ethicists would perhaps agree that morality could not consist of facts or properties that (in J. L. Mackie's phrase - see his Ethics: Inventing Right and Wrong) are part of 'the fabric of the world'. That's to say, few people would argue that morality consists of facts or properties that are our there in the world and detectable by any human being regardless of their sensibility. But maybe there's room for the idea of moral facts and properties on a metphysically more modest construal. Compare social facts, such as legal facts. Surely the law exists? and, in any given jurisdiction, it can be a fact that it's, for instance, illegal to steal, or part your car on the pavement, or whatever. These are objective matters about which one can be right or wrong (if one believes that it's lawful to steal, one is making a mistake).

This is just one model of morality that would render it an objective enterprise about which one can make mistakes, but without any pretensions to moral facts being independent of a certain context or point of view. Now, to return to the suggestion you made about whether morality might be an artefact of rational self-motivation, it seems to me that it could be, so long as one construed rational self-motivation in a way that made co-operative practices the produce of such rational self-motivation. Indeed, some philosophers have argued just that - see, for instance, David Gauthier, Morals By Agreement.

Hope that helps.