Politics are the divine science, after all. (John Adams to James Warren, 1780)
The science of government is my duty to study, more than all other sciences; the arts of legislation and administration and negotiation ought to take place of, indeed to exclude, in a manner, all other arts. I must study politics and war, that my sons may have liberty to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce, and agriculture, in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain. (John Adams to Abigail Adams, 1780)
Jon Rowe has posted here a series of quotes to persuade us that the founders intended the new America to be a religious, but not explicitly Christian nation. The first three of his quotes are from John Adams, a man who made a lifelong mission of studying politics throughout history (see above): what societies succeeded, and what qualities did they have that enabled them to succeed?
Jon's quotes get at part of Adams' answer: religion matters. Any religion is better than no religion, and many religions have historically been adequate to the task in their times and places. But the diverse historical answer to "what has worked somewhere somewhen?" does not imply "anything goes" as the answer to the question "what shall we have as civic religion for America?" The first question was about what religions have been good enough, at least once; the latter question is about what religion is best for the task in general, or at least best for America in particular.
The founders were not, after all, aiming for mere adequacy, or mediocrity, or something which might succeed with a little luck.
So what was John Adams' answer? He, unlike many other founders, wrote out his theory of good government in the form of a state constitution (the state governments were the specific government, supplemented by the union's general government). Here is where John applied what he had learned in his studies of government and religion through the ages, written in the same year as the letters quoted above:
1780 Constitution of Massachusetts, by John Adams (mainly), with the assistance of Samuel Adams and James Bowdoin
As the happiness of a people, and the good order and preservation of civil government, essentially depend upon piety, religion and morality; and as these cannot be generally diffused through a community, but by the institution of the public worship of God, and of public instructions in piety, religion and morality: Therefore, to promote their happiness and to secure the good order and preservation of their government, the people of this commonwealth have a right to invest their legislature with power to authorize and require, and the legislature shall, from time to time, authorize and require, the several towns, parishes, precincts, and other bodies politic, or religious societies, to make suitable provision, at their own expense, for the institution of the public worship of God, and for the support and maintenance of public Protestant teachers of piety, religion and morality, in all cases where such provision shall not be made voluntarily.
And the people of this commonwealth have also a right to, and do, invest their legislature with authority to enjoin upon all the subjects an attendance upon the instructions of the public teachers aforesaid, at stated times and seasons, if there be any on whose instructions they can conscientiously and conveniently attend.
Provided, notwithstanding, that the several towns, parishes, precincts, and other bodies politic, or religious societies, shall, at all times, have the exclusive right of electing their public teachers, and of contracting with them for their support and maintenance.
And all moneys paid by the subject to the support of public worship, and of the public teachers aforesaid, shall, if he require it, be uniformly applied to the support of the public teacher or teachers of his own religious sect or denomination, provided there be any on whose instructions he attends; otherwise it may be paid towards the support of the teacher or teachers of the parish or precinct in which the said moneys are raised.
Any every denomination of Christians, demeaning themselves peaceably, and as good subjects of the commonwealth, shall be equally under the protection of the law: and no subordination of any one sect or denomination to another shall ever be established by law.
How can the same man who wrote Jon's quotes also write this? Easily. Adams knew that virtue sufficient for civil society was found in men of many faiths, and he arguably even would have called all such men "Christian" regardless of faith, e.g. a pious Muslim who was a model of civic virtue might be pleasing to God, and saved, and therefore living the life Christ intended (thereby qualifying as Christian), despite never hearing of, or believing in, Christ.
This is not nearly the same thing as believing that a society that inculcates Islam in its citizens has the same prospects of success and virtue as a neighboring society that inculcates Christianity. One religion is better than the rest, at least on Adams' view, and that was the religion intended by Adams for America. It was a broad vision of Christianity that embraced Unitarianism (then based at Harvard and surging in Massachusetts) but probably not Universalism (or Catholicism).
That Adams thought Christianity superior:
The Christian religion is, above all the religions that ever prevailed or existed in ancient or modern times, the religion of wisdom, virtue, equity and humanity, let the Blackguard Paine say what he will. (John Adams diary entry July 26, 1796)
That Adams might think a pious Muslim was a Christian:
Howl, Snarl, bite, Ye Calvinistick! Ye Athanasian Divines, if You will. Ye will say, I am no Christian: I say Ye are no Christians: and there the Account is ballanced. Yet I believe all the honest men among you, are Christians in my Sense of the Word. (Adams to Jefferson, 1818)
I'll admit I'm stretching the point here - to say that an honest Trinitarian is a Christian is not nearly the same as saying that an honest Muslim is a Christian, but it is a point that leans Jon's way without contradicting my thesis above, so I'm happy to concede it. Besides, even if Adams didn't believe that a pious Muslim was a Christian, it is quite possible that Jefferson, to whom the letter was addressed, did believe it.
Again, though, with respect to Adams (and Jefferson, and Franklin for that matter), the important question is not "who is a Christian?" or "who is virtuous?", but "what is the surest means of inculcating virtue (or Christianity) in the nation?" To this latter question, the answer of the founders' generation was Christianity (and specifically broadly interpreted Protestantism).
Sidebar on article 3 above: note the point that not only is the legislature permitted to require establishment of Protestant instruction, the legislature is required to do so, and it is the right of the people to have their legislature do so! This is not some Orwellian absurdity, this is New England collective covenant communitarianism: I have the right to have Protestantism taught to you at our expense, and you have the same right to have it taught to me.
It's been a while since Jon's and my exchanges on covenant theology, but this was the Massachusetts (Bay colony) way, that we are all bound together so that my well-being depends on your piousness, and vice versa. The original communitarianism was spiritual (your impiety threatens my salvation), the later communitarianism was socioeconomic (your impiety threatens my person and property - as explained by the Theophilus Parsons of the Mass SC). But the theory is the same.