This article from Colonial Williamsburg shows that while deism was influential on the American Founding, there is much more to the story. Some highlights:
Because deism was never a formal sect and had no hierarchy to fix its principles, each adherent could bend it to individual liking, often moving it along traditional Christianity continuum as one manipulates a pointer along a slide rule. On one end of the scale, Washington could be ranked as a deist who attended church services, was a vestryman, quoted the Bible, commended religion, and prayed. At the other end stood Thomas Paine, who recorded his disdain for Christianity in The Age of Reason. In his view, God never communicated with men, Christianity was a fable, and miracles were fictions. Nonetheless, he said he was not an atheist. In a letter to Samuel Adams, Paine wrote that he penned The Age of Reason to keep the French from “running headlong into Atheism” during their revolution, and to ensure that they would hew to “the first article . . . of every man’s Creed who has any creed at all—‘I believe in God.’”
Colonial Williamsburg historian Linda Rowe said, “Franklin, Washington, and Jefferson all attended church services frequently to the end of their lives. They gave money to church building funds of several denominations, and attended Baptist, Presbyterian, Episcopalian, and Unitarian houses of worship. Most made no secret of their conviction that regular religious practice was necessary to public virtue upon which the survival of the republic depended.”
Franklin, whose life almost spanned the eighteenth century, mutated from defining himself as a deist to saying that deism had “perverted” his friends. In his forties, Franklin commended “the excellency of the Christian religion above all others ancient and modern.” As a senior citizen at the Constitutional Convention in Philadelphia, he suggested in vain that the participants pray for God’s guidance. “The longer I live,” he said, “the more convincing proofs I see of this truth, that God governs the affairs of men.” The same Jefferson who clipped the miracles from the New Testament also said, “I am a Christian, in the only sense in which” Jesus “wanted anyone to be: sincerely attached to his doctrines, in preference to all others; ascribing to himself every human excellence; and believing he never claimed any other.”
....Rowe said that “the appeal of moderate deism for Franklin, Washington, Adams, Jefferson, and others was that it allowed for belief in a God who deserved to be worshipped by human beings but was not dependent upon biblical revelation, prophecy, and miracles as matters of faith. I say ‘moderate deism’ because none of these men completely abandoned the religious lessons upon which they were raised. They could be very critical of organized religion and theological twists and turns precisely because, in their view, these man-made institutions had corrupted the simple message of Jesus and obscured the best way to follow his teachings.”...
To some modern historians, “deism” is too simple a term to describe the complexity of the belief system of some Americans in the 1700s. The Reverend Thomas Buckley, SJ, professor of American religious history at the Jesuit School of Theology at Berkeley, said that belief is more complex than saying a person shrugged off Christianity and put on deism: “Too many historians keep calling the ‘founding fathers’ Deists. Even Thomas Jefferson on his worst days was not one of their ranks.” A better term than deist for most of them, Buckley said, is “rationalistic Christian.” Agreement on that point comes from Dreisbach, who said he thinks “there were relatively few Deists in America. There were a few elites who gravitated to a form of theistic rationalism, but we’re talking about a relatively few, albeit influential, elites.”