Hailing from secular Britain I wasn't aware of the distinction. Affirmation actually sounds more religious to me. I'd never particularly associated the idea of an oath with religion but I can see how such an association could sour one on the word 'oath'.

Well obviously the idea of an oath only has value if it is credible, that is why there are often strong cultural taboos against oath breaking. In times past there were often harsh punishments for oath breaking to provide additional enforcement but it is true that in the modern world much of the function of oaths has been transferred to the legal system. Traditionally one of the things that defined a profession was the expectation that members of the profession held themselves to a standard above and beyond the minimum enforced by law however. Professional oaths are part of that tradition, as is the idea of an oath sworn by civil servants and other government employees. This general concept is not unique to the US or to government workers.