The Founding Fathers agreed that government -- at least at the federal level -- shouldn't be able to impose formal religious tests. See Art. VI, Cl. 3 of the US Constitution. But what of private religious tests? (I.e. I want to vote for the "Christian" candidate.) They are permitted in the sense that the voter is allowed to vote for whomever s/he wants, for whatever reason.
John Jay has an oft-repeated quotation that encourages private religious test in favor of "Christians."
“Providence has given to our people the choice of their rulers, and it is the duty, as well as the privilege and interest of our Christian nation, to select and prefer Christians for their rulers.”
Yet, what turned out to be minimal "Christian" standards required for public vetting -- for instance, the "Christianity" of the first half dozen Presidents, perhaps the majority of American Presidents -- was a formal or nominal affiliation with a Christian Church and identification with the Christian label. That's it.
Since Washington's Presidency, there has been no successful "precedent" for privately vetting a Presidential candidate's "Christianity," with a strict confession of orthodox faith (i.e., "Do you George Washington believe in a Triune God? Do you believe the Bible the inerrant, infallible Word of God?), even though many orthodox (Timothy Dwight, William Linn, Jedidiah Morse) wished there were.
The republican form of federal elections that the Founding Fathers established almost landed someone as bad as Aaron Burr in the Presidency.
Washington started many informal Presidential precedents, one of them was religious aloofness, or trying to be all things to all people when pinned down, religiously.
So the "Christianity" of the first four or five Presidents turned out to be, in principle, not much different than the Roman Catholicism of Jack Kennedy, the Quakerism of Richard Nixon, the Southern Baptistism of Bill Clinton, and the unorthodox Christianity of Barack Obama. Jimmy Carter and GWBush stand as the most "orthodox" of Christian Presidents in the modern era, hence the most "Christian" of modern Presidents.
(Reagan? He certainly believed in Providence and thought himself a "Christian." However, it's not clear where he stood on the Trinity, Atonement, Jesus as personal savior. Perhaps someone can enlighten me.)
Perhaps it's better that American Presidents and politicians aren't subject to effective private religious tests like John Jay suggested. It would likely burn too many of them. Orthodox theologians don't necessarily make for the most effective American politicians (see Carter and Bush). By avoiding specific confessions of faith, American federal politicians are effectively shielded from "heresy hunters."
Even the great "John Jay" has given rope with which the heresy hunters could hang him. Though Jay is conceded as one of the "authentic" orthodox Christian notable Founders, one could argue Jay may not have been a "Christian." Or at least that he doubted his Christianity and flirted with Arianism.
As Jay noted in a private letter:
"It appeared to me that the Trinity was a Fact fully revealed and substantiated, but that the quo modo was incomprehensible by human Ingenuity. According to sundry Creeds, the divine Being whom we denominate the second Person in the Trinity had before all worlds been so generated or begotten by the first Person in the Trinity, as to be his coeval, coequal and coeternal Son. For proof of this I searched the Scriptures diligently -- but without Success. I therefore consider the Position of being at least of questionable Orthodoxy."
-- John Jay to Samuel Miller, February 18, 1822. Jay Papers, Columbia University Library.
Do you think he would feel comfortable with a Trinitarian confession of faith for public office?