As a necessary aside, most American probably believe there's no "religious test" to hold public office in America.

After all, the First Amendment bars Congress from making any law regarding an "establishment of religion."

The phrase had a specific meaning to the Founding Fathers. In England, it all started in the 1530s, when Henry the Eighth was having his little disagreement with the Vatican over the Pope's intransigence regarding divorce. (Actually, if you were rich or powerful enough, Rome was fairly liberal in granting "annulments" – rulings, often absurd on their face, that a marriage was void because it had never been "consummated." But Henry's serial polygamies would have tried the patience of anyone -– up through the middle of the 20th Century, anyway, when Western cultures decided fathers were optional.)

By 1714, when Britain's ruling elite went looking for an obscure German prince to take the throne rather than elevate any of the late Queen Anne's Catholic relatives, the Anglican or Episcopal version of the Protestant church was firmly "established" in England.