When should an accountant place his or her duty to the public ahead of his or her duty to a client ...

Because the public (in the form of future shareholders, lenders, customers etc.) is the ultimate user of some services such as the attest function and external corporate reports, and if internal reports or systems are faulty the decisions made will ultimately affect such external stakeholders as well. Focusing just on existing internal stakeholders tends to obscure this. Also when the public hears of a bad decision/action of an accountant, whether it is internal or external doesn't matter; it darkens the reputation of the entire profession as happened in the Arthur Andersen case.