If it came down to the narrowest as you described, if the "one written scripture" we relied upon was from the Judaeo-Christian tradition, we'd still stand a good chance of being "better off than Iran" since we wouldn't be relying upon a book full of distortions of what is within what I myself recognize as Holy Writ, but instead, being from within that book (rather than the Koran, Hadith, Sunnah, or other Islamic writings) be much closer to that exact book of writings (the Bible) which has long served as a significant influence towards Western civilization's entire outlook of what is Truth.
I have it on highest authority (not simply my own opinion, or just what some man or group/organization of men have said) that what I know to be, and what is widely & generally recognized as "the Bible" is true.
The book contains truth.
That does not mean it need be taken entirely literally in wooden two-dimensional fashion throughout. For example; such as what we know as "the world" having been as it is written created in six days does not necessarily equate to that meaning only six 24 hour calendar days, as we otherwise think of "days" to consist of.
There are other ways to grasp the meaning there, without changing or else losing the meaning, yet setting aside counting those six days from merely human perspective. There is another perspective involved, which place of perspective belongs to the One who it is alleged was doing the talking.
I'll give you three guesses whose perspective that is -- but the first two guesses don't count. :^')
A Theocracy, under any name, Iran being the most prominent current, is exactly what the founders wanted to avoid.
Of course some were more hostile to the idea than others, Adams for example. But let there be no mistake, Jefferson and many of the founders were Free Masons first and foremost, the square of such was fundamental to their beliefs, and even much of our government structure was based upon those tenets.
Central to that system is the belief in God. To be more specific, in Free Masonry that would be the God of Abraham, nothing more, nothing less, but let us be realistic, especially in dealing with Jeffersons views, he would not have wanted influence from the other path of the God of Abraham, which was from the Mohammed, and he had not only the historical relevance of dealing with that group (the studied and written history of Masonry going back to the Crusades, Knights Templars etc.), but also a pragmatic personal perspective that came from dealing with the Barbaries.
Again, one can take any individual founder figure, and read their writings. The Federalist Papers read like a soap opera, with each individual have a strong belief, many times diametrically opposed to others.
Ben Franklin was much more pragmatic, and was NOT held in high regard by many of the Christian churches in the area because of his belief that the Church held no sway over government, he was not inclined to submit to the idea that a priest or pastor or whatever hold power in our government, but indeed those people come to the table of the government with all of their wisdom, BUT while at the table of government, they were representative of ALL of their people, not just those of their congregation. In other words, the were public servants while in court, and happily, priest or pastor on their own time.
One of Franklin’s famous quotes which was printed in his Almanac was “Lighthouses have more value than Church’s”. This did nothing to endear him to the hearts of the clergy.
Then we have Adams who was full swing against any sort of religious influence, but even though he thought he was quite clever, one cannot merely make rules for a better life out of thin air, without asking why those rules are in place. They are there because of the Ten Commandments, or perhaps as he argued, it was human nature to be lawful. Most people digress that notion.
In Madison’s writings of the 1820’s (which were considered to be summaries of his early thoughts), he expounded: “And I have no doubt that every new example will succeed, as every past one has done, in shewing that religion & Govt will both exist in greater purity, the less they are mixed together.” and “Every new and successful example of a perfect separation between ecclesiastical and civil matters is of importance.”
But in all, the founders were in agreement that ALL would have a seat at the table, the deist and the pious, and all would have an equal voice.
The main idea in all though, was that after both a bloody revolution with the King of England and the Church of England (both ruled by the same), our nation would not be allowed to follow the same path.