But let me try to explain it in a way different from what I have before:
America's Founding era political-theological landscape, good Protestants they were, was quite fractured and divided. Trinitarianism, though associated with the essentials of "Christianity" by some back then (and today, think of CS Lewis' "Mere Christianity"), became a part of the disputed definition of "Christianity," not an essential doctrine of the "general Christianity" that may have united them. The unitarians by that time had been keeping their mouths shut on those doctrines whenever they mentioned God; they got used to speaking in more generally theistic terms so as not to offend their unitarian private convictions or the convictions of the orthodox. In short they were the ones, out of personal necessity, who got really good at lowering the common denominator of God talk. They even realized lowering the denominator of God talk and combining it with a natural theology with which all "reasonable" men could agree enabled them to communicate metaphysical-theological truths with, among others unconverted Native Americans and even Muslims.
In short, the unitarians made for the best neutral referees among bitterly divided sectarian dogmatists and were most suited for uniting a country divided by theological dogma. We know that J. Adams and Jefferson turned out to be self conscious unitarians. And George Washington and James Madison were closeted about their Christological opinions and systematically spoke in more generally theistic uniting God terms. I wonder why.
And so we have John Adams' 1798 Thanksgiving Proclaiming that sounded Trinitarian on its surface, but may have also been consistent with pious unitarian consciences (most of whom believed God is Father, Jesus, though not fully God, was Redeemer, and also found some way to explain the existence of the Holy Spirit without believing in the Trinity).
John Adams regretted that because...surprise surprise, it was too sectarian. Its Trinitarian sounding surface was too easily mistaken for Presbyterianism. And sectarian power at the top was a slippery slope towards persecution. At least, that's how John Adams saw it. John Adams' June 12. 1812 letter to Benjamin Rush explains this AND the letter is also the source of Adams' Trinity mocking quote ("The Trinity was carried in a general Council by one vote against a Quaternity...") that I have oft-quoted.
Adams comes off as a ninny, perhaps half drunk, paranoid about religious persecution. But it was that paranoia about NOT sounding sectarian that helped unite the American Founding era under a more general God. I'll let you read the entire letter for its context. Here's a BIG taste:
The next paragraph requires a graver answer. But a Volume would not suffice. Take a hint. I have lived among Infidel Philosophers more than half a Century, and been engaged in continued disputes with them. This has compelled me to spend more time in reading Universal History but especially Ecclesiastical History, than has been for my Interest or Comfort. While the Result has been an increasing Love for Christianity, as I understand it, a growing Jealousy of the Priesthood has accompanied it all the way. Levites, Magi, Faquirs, Mandarines, Mufti, Druids, Popes; Cardinals, Archbishops, Bishops, Bernardines, Jacobins, Dominicans, Westleys, the Prophet of Wabash, or Tippecanoe, Nimrod Hughs, Christopher McPherson, and even Priestly and Price, even Dr. Ewing, Dr. Rogers and Dr. Dwight have conspired together to rivet to my soul the Duty and Necessity of Tolleration.
These general assemblies of Presbyterian Divines are general Councils in embrio. We shall have Creeds and Confessions, Church discipline and Excommunication. We shall have, the civil Government overawed and become a Tool. We shall have Armies and their Commanders under the orders of Monks. We shall have Hermits, commanding Napoleons, I agree with you, there is a Germ of Religion in human nature so strong that whenever; an order of Men can persuade the People by, flattery or Terror, that they can have salvation at their disposal, there can be no end to fraud Violence or Usurpation. Ecumenical Councils produce Ecumenical Bishops and both subservient Armies, Emperors and Kings.
The National Fast recommended by me turned me out of office. It was connected with the general assembly of the Presbyterian church, which I had no concern in. That assembly has alarmed and alienated Quakers, Anabaptists, Mennonists, Moravians, Swedenborgians, Methodists, Catholicks, protestant Episcopalians, Arians, Socinians, Armenians, &c,&c,&c, Atheists and Deists might be added. A general Suspicion prevailed that the Presbyterian Church was ambitious and aimed at an Establishment as a National Church. I was represented as a Presbyterian and at the head of this political and ecclesiastical Project. The secret whispers ran through them [all the sects] "Let us have Jefferson, Madison, Burr, any body, whether they be Philosophers, Deists, or even Atheists, rather than a Presbyterian President" This principle is at the bottom of the unpopularity of national Fasts and Thanksgiving. Nothing is more dreaded than the National Government meddling with Religion. This wild Letter, I very much fear, contains seeds of an Ecclesiastical History of the U.S. for a Century to come.
Okay. Now get ready for the CONTEXT of John Adams' The Trinity prevailed by one vote against a Quaternity quote. The context is close tie breaking elections illustrating the nature of groups of men to divide themselves, intractably.
The similitude between 1773 and 1774, and 1811 and 1812 is obvious. It is now said by the Tories that we were unanimous in 1774. Nothing can be farther from the Truth. We were more divided in 74 than we are now. The Majorities in Congress in 74 on all the essential points and Principles of the Declaration of Rights, were only one, two or three. Indeed all the great critical questions about Men and Measures from 1774 to 1778 were decided by the vote of a single state, and that vote was often decided by a single Individual. Jumble and Chaos as this Nation appears at this moment, I never knew it better united. It is always so. The History of the World is nothing but a narrative of such divisions. The Stuarts abdicated or were turned out and William came in by one or two votes. I was turned out by the votes of S. Carolina not fairly obtained. Jefferson came in by one vote, after 37 Tryals between him and Burr. Our expedition against Cape Breton and consequent Conquest of Louisburg in 1745 which gave peace to the World was carried in our House of Representatives of Massachusetts by one single vote. The abolition of old Tenor in 1750 was decided by one vote. What is more awful than all. The Trinity was carried in a general Council by one vote against a Quaternity: the Virgin Mary lost an equality with the Father, Son and Spirit only by a single suffrage. All the great affairs of the world temporal and spiritual, as far as Men are concerned in the discussion and decision of them are determined by small Majorities. The Repulsion in human nature is stronger than the Attraction. Division, Separation are inevitable.
Adams then starts in with a diva-like discussion of "Boudoirs." Adams seems self consciously aware of his "feminine side" in this letter. That's probably when the alcohol fully kicked in.