Could the presumption of innocence protect the guilty?

At criminal trial, we demand that those accused of criminal wrongdoing be presumed innocent until proven guilty beyond any reasonable doubt. What are the moral and/or political grounds of this demand? One popular and natural answer to this question focuses on the moral badness or wrongness of convicting and punishing innocent persons, which I call the direct moral grounding. In this essay, I suggest that this direct moral grounding, if accepted, may well have important ramifications for other areas of the criminal justice process, and in particular those parts in which we (through our legislatures and judges) decide how much punishment to distribute to guilty persons. If, as the direct moral grounding suggests, we should prefer under-punishment to over-punishment under conditions of uncertainty, due to the moral seriousness of errors which inappropriately punish persons, then we should also prefer erring on the side of under-punishment when considering how much to punish those who may justly be punished. Some objections to this line of thinking are considered.