A recent book argues that many of the key Founding Fathers were neither Deists - if Deism is the ad hoc view that God exists just to start up the universe - nor Christian - rejecting as incoherent the defining doctrines of the Trinity, the Incarnation, and substitutionary atonement. Rather, they embraced classical theism including the dependence of the world on God's concurrence and even prayed.
This is probably true, and useful in clarifying current debates over homosexuality, abortion, and religious liberty, by bearing an intellectual witness to the possibility of such a position. In part this position was encouraged from within the heart of the church itself by distinguishing natural theology (what can be known of God by our natural lights apart from special revelation, which served as the perambulatory articles to sacred theology) and sacred, specially revealed theology. According to the church, the world prior to Christianity had this knowledge exhibited substantially in many philosophers apart from the church.
Such knowledge included knowledge of God, of man, and of morality. It served to reveal or need of extraordinary mercy and thus at best prepared the heart and mind to receive revelation if God should provide it. At least, the awareness of it acted as a curb on the selfish tendencies of humans and kept many of them from being as bad as possible. It also informed public judgements and made possible states and governments to oversee the public peace.
The rational theism and Unitarian faith of the Fathers is simply a taking on the natural theology of the church and the ancients and foregoing the sacred doctrines. Such a position is tolerant of religion in the classical form precisely because of so much common ground with it. Also, the Church recognized the existence of natural theology, would certainly appreciate that if one could not freely accept Christianity that they be encouraged to live by natural law. The church is distinct from the state but both recognize the same God, albeit from their own respective spheres.
This was illustrated in the debate between Ryan and Biden. Ryan is coherent in affirming the inseparability of his religious life from his political life since the both have an ultimate goal in common, service to God while also affirming that the rationale for his support for pro-life is separable from his Church beliefs such that a non-churched rational person can hold them and a public person can advocate them as laws. Biden on the other hand is incoherent in claiming that abortion is an exclusively church based belief that he doesn't want to impose on others who are not in his church. His Church holds that his pro-life view is that a pro-life stance is only a Christian belief because it is true prior to Christian belief. And thus justice for the unborn is an obligation to all. (One notes in passing that Catholics like Biden often argue that they must advocate left leaning policies because that's what Jesus would do.)
Natural or philosophical theism is thus a public position. There are arguments that metaphorically demonstrate theism that depend on premises that are not obviously false. Further, theism is arguably a necessary practical postulate for statesmanship. If God exists, then that man has a specific nature that makes humans ends as such, then there us nothing puzzling about all humans all the time having a right to life or that there is a purpose to humans being make and female, that stands true apart from church doctrine. Just as the Church appreciates the natural theistic politician, that politician can appreciate the church which treats the dictates of natural theology as God's moral law for life, since such dispositions also make for good citizens by the natural theist's point of view.
Thus the preponderance of both Rational and Christian theists in America created an excellent state of affairs for the institution of the separation of Church and State.
It is also the idea of rational autonomy that encourages free market thought while the idea of the social nature of humanity encourages the formation and protection of free institutions of civil society. Conservativism, whether economic or social finds it's roots in the tradition of rational theology, by finding them in the American Founding.
But the triumph of empiricism and it's inevitable consequences of phenomenalism, behaviorism, emotivism, and prescriptive relativism has cut people off from the sources of rational theism. When given the choice between living passively in the world of images or living actively in the world of intelligible substances and persons, people have chosen the former. Based on that choice both the church and the state make no sense as institutions governed by objectively based principles and can only be the result of successful power manipulation -- not that there could be anything wrong with that.
Hence we get statements like "The Tea Party has been taken over by wackos" and meaning by that that there is a preponderance of social conservatives in an ostensively fiscally conservative movement. Whatever the current character of the Tea Party is it's inception was characterized as a spontaneous people movement unified explicitly around two things: de jure, fiscally responsible government and the reservation of the foundational document, the Constitution. The point was to keep the American experiment going. The second thing, de facto, of those who came out to rally for this cause, from the becoming there was a significantly large percentage of them that were regular church attendees. In that sense, the "wackos" were there all along. Even when they were careful to only use the TP microphone for certain fiscal issues they would still advocate for social issues on the side. This is not surprising since with the triumph of empiricism in public thought, the church has been the last safe ground natural theism of the non-Christian Founders.
Empiricists choose to view the world exclusively from a third person perspective rather than a first person perspective, even though there is no incoherence in viewing the world both ways. Consequently, they don't "see" any philosophical basis for marriage rights, free exercise of religion as a life-system rather than a privatized "spirituality", or pro-life arguments. That is, they don't see because the don't look. As a result, they don't even "see" that their own posture is self-reverentially incoherent. The commitment to empiricism must be a priori, if it is rational at all.
This is also why it's so hard for us to get the Founders right. The only groups that defend their secular philosophy are religious groups.