[A note on my use of capitalization; when I talk about theological unitarianism, I use a small "u," when I talk about Church Unitarianism, I use a capital U. Jefferson, Madison and Franklin were unitarians. John Adams was a unitarian and a Unitarian.]
Founding Era unitarianism had a right wing to it, although I wouldn't put John Adams, Thomas Jefferson, or Ben Franklin there. Rather, I'd put John Marshall, Joseph Story, and perhaps Jared Sparks there. By "right wing," I mean, they were more likely to be antidisestablishmentarians; they were likely to support "Christianity" as the only true religion; and they likely thought the scriptures to be infallible. I wouldn't call them theistic rationalists like Jefferson, Adams, and Franklin because it's not clear that the more biblical unitarians thought reason superseded revelation as Jefferson, Adams, and Franklin clearly did. Though men like Marshall, Story, and Sparks nonetheless denied Original Sin, the Trinity, Incarnation, and Atonement and believed men were saved through good works. To them, this is what "Christianity" was all about. In 18th Century America, all of the official Churches termed these beliefs heresy (and lots of elite educated figures privately believed in these heresies). By the 19th Century this "heresy" became, at least in the North East, a socially respectable form of liberal Protestantism.
Check out this link which gives the opinions of three famous historical figures on whether "Christianity" is foundational to America's government. Joseph Story and John Marshall answer in the affirmative, Madison in the negative. What's notable is that all three figures are unitarians. Keep in mind that when John Marshall answers "[t]he American population is entirely Christian, & with us, Christianity & Religion are identified...," he considered unitarianism that denied Original Sin, the Trinity, Incarnation, and Atonement to be "Christian."
Here is an interesting letter from Joseph Story explaining his biblical unitarianism to a Trinitarian critic that argued unitarians were "deistic":
TO WILLIAM WILLIAMS, ESQ.
Washington, March 6th, 1824.
I acknowledge with pleasure your letter of the second of February, which reached me a very few days since. What you say of the false statements in the prints respecting Unitarians does not surprise me; for I well know that bigotry, and misapprehension, and ignorance are very like to lead men to the most extravagant opinions. The Unitarians are universally steadfast, sincere, and earnest Christians.
They all believe in the divine mission of Christ, the credibility and authenticity of the Bible, the miracles wrought by our Saviour and his apostles, and the efficacy of his precepts to lead men to salvation. They consider the Scriptures the true rule of faith, and the sure foundation of immortality. In short, their belief is as complete of the divine authority of the Scriptures, as that of any other class of Christians.
It is a most gross calumny, therefore, to accuse them of treating the Bible and its doctrines as delusions and falsehoods, or of an union with Deists. In sincere unaffected piety, they yield to no persons. They differ among themselves as to the nature of our Saviour, but they all agree that he was the special messenger of God, and that what he taught is of Divine authority. In truth, they principally differ from other Christians in disbelieving the Trinity, for they think Christ was not God, but in the Scripture language "the Son of God."
I think it not impossible that Deists may look upon them with more favor than upon other Christians, because they have confidence in human reason as a guide to the interpretation of the Scriptures, and they profess what the Deists consider more rational and consistent opinions than the Calvinists. But beyond this, I believe, that the Deists have no kindness for them, and as to connection with them, it is an utter absurdity. You do the Unitarians, therefore, no more than the justice which I should expect from your liberality, in disbelieving such tales. But I will not trouble you any more with this controversial subject. I should exceedingly rejoice to see you again in New England, where you would see them as they are, and you would find, that, although changes of opinion may have occurred, a strong religious feeling and a spirit of improvement universally prevail.
May you long, my dear sir, enjoy the happiness that results from a pure life and elevated pursuit, This is the wish of your most obliged friend,
Here is the problem for Story: Perhaps his Trinitarian critic's views were more accurate than what Story gives him credit for. Story erred when he asserted all Unitarians were biblical, perhaps wishful thinking on his part. Let us not forget that Jefferson and Adams embraced the label "unitarian" along with Story et al. Jefferson used his reason to edit those parts of the Bible with which he disagreed. And Adams elevated reason so far over revelation that he explained even if God revealed the doctrine of the Trinity to him with Moses at Mt. Sinai, he still wouldn't believe it because man's reason proves 1+1+1 = 3, not 1. As he wrote to Jefferson, Sept. 14, 1813:
The human understanding is a revelation from its maker which can never be disputed or doubted. There can be no skepticism, incredulity, or infidelity here. No prophecies, no miracles are necessary to prove the celestial communication.
This revelation had made it certain that two and one make three, and that one is not three nor can three be one. We can never be so certain of any prophecy, or the fulfillment of any prophecy, or of any miracle, or the design of any miracle, as we are from the revelation of nature, that is, nature’s God, that two and two are equal to four. Miracles or prophecies might frighten us out of our wits, might scare us to death, might induce us to lie, to say that we believe that two and two make five, but we should not believe it; we should know the contrary.
Had you and I been forty days with Moses on Mount Sinai, and admitted to behold the divine glory, and there been told that one was three and three one, we might not have had the courage to deny it, but we could not have believed it.
As noted these unitarians believed their religion to be a form of liberal rational Christianity. If, as Marshall and Story believed, Christianity had some type of organic connection to government and if their creed which denied original sin, the Trinity, Incarnation, and Atonement, and Jefferson's and Adams' "Christianity" which denied the infallibility of the Bible and thought reason could edit the irrational parts of scripture all qualified as "Christianity" with which government had some kind of special relationship, we should easily be able to see how this could unleash undesirable theological disputes (i.e., "no, government should promote only true Christianity, not your heresy!"). This is exactly what Madison had in mind when he remonstrated against Patrick Henry's Bill to support teachers of the Christian religion; he didn't want the law to have to decide what is Christianity.
In the modern era we see similar disputes with the "Mormon Christian" Mitt Romney running for President and evangelicals asserting he really isn't a Christian. As Romney supporter Hugh Hewitt admonished: "But leave questions about theology --about revelation-- out of politics." This is exactly as America's Founders believed. In order to solve the problem of sectarian disputes, revealed religion had to be driven from politics. Although they did appeal to a natural theology, a theology of reason governed by a generic undefined deity: "Nature's God," or "Providence."