John Adams published a wonderfully short and readable guide to constitution-making entitled "Thoughts on Government" in 1776, coinciding with a recommendation from Congress that the colonies should establish new governments. This essay of Adams' is a window into his thinking on why government should have the forms he was advocating. It is a short read, and well worth the effort.
Before getting into the details of government, Adams lays out an important point, perhaps easily overlooked today: the purpose of good government is to foster good society, defined as virtuous society, and this is clearly distinguished from a society of virtuous people. Adams puts it as follows:
"We ought to consider what is the end of government, before we determine which is the best form. Upon this point all speculative politicians will agree, that the happiness of society is the end of government, as all divines and moral philosophers will agree that the happiness of the individual is the end of man... [and] All sober inquirers after truth, ancient and modern, pagan and Christian, have declared that the happiness of man, as well as his dignity, consists in virtue. Confucius, Zoroaster, Socrates, Mahomet, not to mention authorities really sacred, have agreed in this."
Two points to take from this are:  Adams is reifying society, and specifically anthropomorphizing it (society is a real thing distinct from its members, and it has the same sorts of qualities that people have); and  happiness as a goal of people (and societies) is not what a casual reader might think is happiness (the philosophers' technical term for the casual sort of happiness is "contentment"), but rather happiness is virtue, or moral goodness.
Government does not exist to make individual people happy/virtuous; that is what mankind in general exists for ("the happiness of the individual is the end of man"). Instead, government exists for the complementary purpose of making society happy/virtuous ("the happiness of society is the end of government").
This matters because it addresses some of the foundation-level questions of the Christian-Nation hypothesis, such as "what on earth can it mean for a nation to be Christian?". An answer would be that if a nation (society) is an anthropomorphic entity, then it is a person capable of being Christian, or at least capable of acting as a Christian. In other words a nation is capable of humility, charity, forgiveness, contrition, and all the rest.
This also matters, of course, because of Jefferson's easily misunderstood phrase "pursuit of happiness" in the DoI, which readers in the founder's era would have understood as pursuit of virtue (self-development, moral improvement) rather than pursuit of entertainment or diversions. God didn’t give us a fundamental right to seek entertainment; or at least few in the founder’s era would have thought so. This is not to say that Jefferson agreed with Adams on everything, only that he used terms like “happiness” in the same way as Adams did, so that Jefferson and Adams could converse on these subjects without talking past one another.
Anyway, what has this to do with a Christian nation, rather than a merely anthropomorphic nation? Only this, that the founders would have taken the same approach to coercion of the conscience of a nation as they took to coercion of the conscience of individuals. As I have argued before, the Christianity of the founding era placed heavy emphasis and expended great theological energy on things we don’t think too deeply about anymore, and one of those things was non-coercion in conversion.
For a nation to be truly Christian it must first be truly free, and then in freedom choose to be Christian. If the constitution of the government of the nation is in any way an embodiment of the conscience of the nation, then it must not be hard-wired in any Christian sense, for to do so would preclude the conversion experience necessary for true Christianity. For the nation to be free it is also necessary, but not sufficient, that the government be well administered, and persistent good administration (generation after generation) was Adams’ pragmatic goal for the rest of the essay (“Nothing is more certain, from the history of nations and nature of man, than that some forms of government are better fitted for being well administered than others.”)
Anyway, there are all sorts of other gems in the essay, such as this swipe at English political philosophy: “A man must be indifferent to the sneers of modern Englishmen, to mention in their company the names of Sidney, Harrington, Locke, Milton, Nedham, Neville, Burnet, and Hoadly. No small fortitude is necessary to confess that one has read them. The wretched condition of this country, however, for ten or fifteen years past, has frequently reminded me of their principles and reasonings. They will convince any candid mind, that there is no good government but what is republican. That the only valuable part of the British constitution is so; …”
Another is this radical thought on what we now call second-amendment issues: “A militia law, requiring all men, or with very few exceptions besides cases of conscience, to be provided with arms and ammunition…”. This goes beyond a right to keep and bear arms, it recommends an entitlement to arms and ammunition.
Adams’ ending sums up his anticipated effects paternalistically: “A constitution founded on these principles introduces knowledge among the people, and inspires them with a conscious dignity becoming freemen; a general emulation takes place, which causes good humor, sociability, good manners, and good morals to be general. That elevation of sentiment inspired by such a government, makes the common people brave and enterprising. That ambition which is inspired by it makes them sober, industrious, and frugal.” The point, quite clearly, is not freedom for freedom’s sake, but rather freedom to make people morally better. So, as an end, the purpose of good government is happiness/virtue of society, and in pursuing that end a fortuitous effect will also be happiness/virtue of individuals.