This thesis is controversial to both sides. The hard core secularists like Christopher Hitchens and Sam Harris attack "religion" in general; they don't signal out Christianity and don't want America's Founders to have appreciated the variety of religions they criticize. They want them to be atheists or strict Deists along the lines of Thomas Paine who thought all of these religious systems to be "no good."
Traditional Christians on the other hand want orthodox Christianity to "own" the Founding to the exclusion of other religions.
I think of one Christian Nationalist blogger (no dummy, and his two favorite figures are David Barton and D. James Kennedy) who has said things along the lines of the FFs thought government should publicly support "Christianity" not other religions. And then he defined "Christianity" with orthodoxy and noted it would be therefore inappropriate for America to support Mormonism, because it is not "Christianity" but another religion.
If you pin down most other Christian Nationalists, I'm sure you'd be able to get them to make assertions/conclusions like this.
And traditional Christians otherwise have a problem with the notion of "religion in general." Take for instance Thomas West, a conservative scholar for whom I have a great deal of respect. He writes:
There is no such thing as "religion in general." All meaningful government support of religion is always support of a particular religious view, as 19th-century Catholics bitterly experienced. Today, support of "religion in general" would include taxpayer funding of Wiccans, Satanists, Muslims (including those who teach hatred of America), and worshippers of that favorite goddess of some feminists, "Our Sweet Sophia."
On sound theological grounds, West may be right to debunk "religion in general." His problem is 1) the key Founders believed in this illusion, and that 2) that's what they protected in the US Constitution -- "religion" not "Christianity." Now, the entire notion of "natural law/rights" has been, some prominent scholars argue, debunked in a metaphysical sense (Mark Tushnet of Harvard Law says it's like believing "ghosts"). West's prime mission as a scholar is to "vindicate" the Founding against such debunking. So he might want to exercise more caution when he tries to debunk religion in general, because it only serves to debunk what he's trying to vindicate.
Now, I won't try to defend the idea that the Founders included Satanists in their vision for "religion in general." But the following is a list of "religions" which they believed were "sound" and valid ways to God: Orthodox or unorthodox Christianity, Judaism, Islam, certain forms of Deism, Hinduism, Confucianism, Native American spirituality and pagan Greco-Romanism. Putting them together, you certainly get "religion in general" not "Christianity in particular."
“It has pleased the Providence of the first Cause, the Universal Cause, that Abraham should give religion not only to Hebrews but to Christians and Mahomitans, the greatest part of the modern civilized world.”
–- John Adams to M.M. Noah, July 31, 1818.
"θεμις was the goddess of honesty, justice, decency, and right; the wife of Jove, another name for Juno. She presided over all oracles, deliberations, and councils. She commanded all mortals to pray to Jupiter for all lawful benefits and blessings. Now, is not this (so far forth) the essence of Christian devotion? Is not this Christian piety? Is it not an acknowledgment of the existence of a Supreme Being, of his universal Providence, of a righteous administration of the government of the universe? And what can Jews, Christians, or Mahometans do more?"
-- John Adams to Thomas Jefferson, October 4, 1813.
"Where is to be found Theology more orthodox or Phylosophy more profound than in the Introduction to the Shast[r]a [a Hindu Treatise]? 'God is one, creator of all, Universal Sphere, without beginning, without End. God Governs all the Creation by a General Providence, resulting from his eternal designs. — Search not the Essence and the nature of the Eternal, who is one; Your research will be vain and presumptuous. It is enough that, day by day, and night by night, You adore his Power, his Wisdom and his Goodness, in his Works.'"
– John Adams to Thomas Jefferson, December 25, 1813.
"Both house and ground were vested in trustees, expressly for the use of any preacher of any religious persuasion who might desire to say something to the people at Philadelphia; the design in building not being to accommodate any particular sect, but the inhabitants in general; so that even if the Mufti of Constantinople were to send a missionary to preach Mohammedanism to us, he would find a pulpit at his service."
-- Ben Franklin, Autobiography.
“Every religion consists of moral precepts, and of dogmas. In the first they all agree. All forbid us to murder, steal, plunder, bear false witness &ca. and these are the articles necessary for the preservation of order, justice, and happiness in society. In their particular dogmas all differ; no two professing the same. These respect vestments, ceremonies, physical opinions, and metaphysical speculations, totally unconnected with morality, and unimportant to the legitimate objects of society. Yet these are the questions on which have hung the bitter schisms of Nazarenes, Socinians, Arians, Athanasians in former times, and now of Trinitarians, Unitarians, Catholics, Lutherans, Calvinists, Methodists, Baptists, Quakers &c. Among the Mahometans we are told that thousands fell victims to the dispute whether the first or second toe of Mahomet was longest; and what blood, how many human lives have the words ‘this do in remembrance of me’ cost the Christian world!…We see good men in all religions, and as many in one as another. It is then a matter of principle with me to avoid disturbing the tranquility of others by the expression of any opinion on the [unimportant points] innocent questions on which we schismatize, and think it enough to hold fast to those moral precepts which are of the essence of Christianity, and of all other religions.”
– Thomas Jefferson to James Fishback, Sept. 27, 1809
“Faith is recommended as a Means of producing Morality: Our Saviour was a Teacher of Morality or Virtue, and they that were deficient and desired to be taught, ought first to believe in him as an able and faithful Teacher. Thus Faith would be a Means of producing Morality, and Morality of Salvation. But that from such Faith alone Salvation may be expected, appears to me to be neither a Christian Doctrine nor a reasonable one….Morality or Virtue is the End, Faith only a Means to obtain that End: And if the End be obtained, it is no matter by what Means.” [Bold mine.]
– Benjamin Franklin, “Dialogue between Two Presbyterians,” April 10, 1735.
The following quotations concern "The Great Spirit" which was the unconverted Natives' specific God term. This is important because the Natives were (as far as I know) the largest population of non-identifactory Christians during Founding era America and as Presidents Washington, Jefferson and Madison all used that term by name suggesting it was the same "Providence" Christians, Jews and Muslims worshipped and thus a valid way to God.
"I now send my best wishes to the Cherokees, and pray the Great spirit to preserve them."
-- George Washington, TALK TO THE CHEROKEE NATION, August 29, 1796.
"I now sincerely wish you a good Journey and hope you may find your [families and] Brothers well on your Return, and that [the Great Spirit above] may long preserve your Nations in peace with each other and with the United States."
-- George Washington, To THE CHIEFS AND WARRIORS, REPRESENTATIVES OF THE WYANDOTS, DELAWARES, SHAWANOES, OTTAWAS, CHIPPEWAS, POTAWATIMES, MIAMIS, EEL RIVER, WEEAS, KICKAPOOS, PIANKASHAWS, AND KASKASKIAS, November 29, 1796.
Note, one of Washington's aides wrote this speech and Washington HIMSELF crossed out the word "God" and wrote in "the Great Spirit above."
"But we thank the Great Spirit who took care of you on the ocean, and brought you safe and in good health to the seat of our great Council; and we hope His care will accompany and protect you, on your journey and return home; and that He will preserve and prosper your nation in all its just pursuits."
-- Thomas Jefferson, to the Choctaw Indians, 1803.
"My children, I thank you for your visit and pray to the Great Spirit who made us all and planted us all in this land to live together like brothers that He will conduct you safely to your homes, and grant you to find your families and your friends in good health."
-- Thomas Jefferson, to the Cherokee Nation, 1806.
"I have a further advice of my Red children. You see how the country of the eighteen fires is filled with people. They increase like the corn they put into the ground. They all have good houses to shelter them from all weathers, good clothes suitable to all seasons; and as for food, of all sorts, you see they have enough and to spare. No man, woman, or child, of the eighteen fires, ever perished of hunger. Compare all this with the condition of the Red people. They are scattered here and there in handfulls. Their lodges are cold, leak, and smoky. They have hard fare, and often not enough of it.
"Why this mighty difference? The reason, my Red children, is plain. The white people breed cattle and sheep. They spin and weave. Their heads and their hands make all the elements and productions of nature useful to them.
"It is in your power to be like them. The ground that feeds one lodge by hunting, would feed a great band by the plough & the hoe. The Great Spirit has given you, like your white brethren, good heads to contrive, and strong arms, and active bodies. Use them like your white brethren of the eighteen fires, and like them, your little sparks will grow into great fires. You will be well fed, dwell in good houses, and enjoy the happiness for which you, like them, were created. These are the words of your father to his red children. The Great Spirit who is the father of us all, approves them. Let them pass through the ear in to the heart. Carry them home to your people; and as long as you remember this visit to your father of the eighteen fires, remember these are his last and best words to you!"
-- James Madison, to the Cherokee Indians in 1812.
Finally, here is a quotation by Benjamin Rush. Rush was a Trinitarian Universalist. The following illustrates that he believed the people were free to choose Christianity as religion which they wanted to see inculcated in America. I suppose it's a soft version of the Christian Nation idea. However, he also notes the concept of religion in general about which I speak, thus simultaneously argues for a soft version of that standard. The content of "sound religion" is not anything particularly Christian (i.e., Christ's atonement) but rather the teaching of the existence of an overriding Providence and future state of rewards and punishments. Note how what I put in bold can connect Christianity with all sorts of exotic world religions (see the quotations from Adams above connecting it to Hinduism and pagan Greco-Romanism).
Such is my veneration for every religion that reveals the attributes of the Deity, or a future state of rewards and punishments, that I had rather see the opinions of Confucius or Mahomed inculcated upon our youth, than see them grow up wholly devoid of a system of religious principles. But the religion I mean to recommend in this place, is that of the New Testament.
-- Benjamin Rush, Of the Mode of Education Proper in a Republic.
Note the negotiability of "Christianity" as the religion which government should support or teach. It is something Rush merely "recommends," not some kind of essentially true religion to the exclusion of false religions that forms the foundation of America's public, organic, principles.
All of this is why I believe the religion spoken of in the Declaration of Independence is a far more generic inclusive natural religion, than anything particularly or exclusively Christian.