I am a fan of Rand's but I only partially agree. Certainly at least some of the founders were religious 'God-fearing' men. Rand's ideas of morality also went far beyond the political areas of the documents.
The founder's acknowledged that Man's rights to life, liberty and the pursuit of happiness were endowments from the "Creator" without referencing any specific religious authority and thereby avoiding conflictiing religious arguments.
Similarly, Ayn Rand used reason to deduce that man's highest value is his own life, and she extended her ethics from that logical base -- while avoiding religion's ethical traps.
Rand points out on page two of "The Objectivist Ethics" that ... "The greatest of all philosophers, Aristotle, did not regard ethics as an exact science; he based his ethical system on observations of what the noble and wise men of his time chose to do, leaving unanswered the questions of: why they chose to do it and why he evaluated them as noble and wise."
Ayn Rand proceeds, in subsequent pages of "The Objectivist Ethics," to fully develop and explain the whys and wherefores of her rational ethics.