No, they do not. However, these writings were composed of men who are mere mortals, humans that is, who breath air and bleed blood, just as you and I. They made a good talk of "freedom, liberty and equality for everyone", but then engaged in perplexing behavior not much different than politicians of today.
The FF's got us where we are today, and they deserve much credit for that, for everyone should be eternally thankful for their sacrifices. But many of their ideas and philosophies are contrary to a free people. And by reading many of their works, they were not really interested in a free society. They were authoritarian. The main difference is that the "governments" control was not nearly as far reaching as it is today. 200-250 years ago, if I did not like what government was doing in the colonies, I could pick up and move west, and essentially be under no government control. Such laws against "witchcraft", sodomy, sexual positions etc would not apply any longer. I think many envisioned an authoritarian theocracy on a small scale, with no idea that as the authoritative nature of government grew, "morals" would decline. And as people realized some of the warped translations of the Bible that were prevalent at that time, they realized such authoritative edicts were contrary to the ideas that the FF's espoused(Not to mention woman's suffrage and slavery). This also resulted in, over the years, many people rejecting true Christianity because of the warped version that they had been blindly following.
What's the point of this rant? To answer the question again, no, the FF's did not intend to remove relgion from the public sphere. Just keep in mind, though, that they were not really for religious freedom, unless you were of some type of "christian religion", and even then, you better not let your teachings stray too far from theirs. Religions of all kinds have every rightful place in the public sphere. Notice I said all religions, not just ones I may agree with. That being said, I agree with the supposed intent of the FF's to not allow a "state sponsored, recognized" religion for "the free excerices thereof" is to not be prohibted.
You've stood the issue on its head. The central government was very small and modestly funded through its first century of existence. During the same period most American professed belief in theism of a Judeo-Christian variety and sought to live the greater part of their lives accordingly.
Central government got its initial boost in the Civil War, but the afterburners weren't lit until the dawn of the 20th Century when the state-based counterfeit millennialism of communism and socialism struggled to displace the personal religion-based salvation and self-reliance of Judeo-Christianity. As a consequence of this tendency, ordinary Americans started to buy into the lie that morality is an individual standard and that God is irrelevant. The more they surrendered to this peculiar anomie, the more powerful central government became.
Central government grows to fill the vacuum left when people reject God and religion-based morality out as regulating forces in their personal lives. They surrender an endoskeleton of religion-based spine and take on a exoskeleton of governmental coercion. In time, God-haters are no longer individual up-standing men but mere clattering social insects.
That is their sure and unavoidable fate. But our Constitution was designed by and for men--not insects.