The First Amendment was designed to do two things: 1) disempower completely and forever the federal government from sticking its nose into religious matters, and 2) to preserve the power of the states and the citizens of those states to do anything they desired in the realm of religion up to and including the establishment of state religions. For over 170 years this is the way we did things in this country: we had a collection of (to use your snide term) "mullah" states.
It wasn't until the 1940s that a liberal SCOTUS employed a taken-out-of-context sentence from an obscure letter by Jefferson to the Danbury Baptists of Massachussets to completely invert the First Amendment from its original meaning. As a result the federal government now expends an enormous amount of time and resources aggressively sticking its nose into matters of religion, having re-interpreted the free exercise clause to require freedom from religion instead of freedom of religion. And states have been wholly stripped of any authority to legislate concerning religion. If you examine state constitutions you will find a great many have religion clauses that have been completely made into nullities because of the federal SCOTUS anti-religion jihad begun in the 1940s.
And I strongly disagree with you on the effect of allowing religious sentiment and moral guidance in government. To suggest that this would turn us into an 8th Centur Islamic state is to ignore the historical truth of the American experience for over 170 years. It is more than hyperbole-it is a lie.
The alternative, which you seem to delight in, is not a just and free society, but one marked by moral relativism, anomie, and reckless personal indulgence to the detriment of society as a whole. Your atheist vision is not of a City on a Hill as envisioned by the founders, but of a Hollyweird hell in a chasm of atomistic creatures inventing and re-inventing their own laws on the fly.
No they didn't.
Let me ask a question; in the 180 +/- years between the signing of the constitution and the USSC ruling on school prayer was our gov. a theocracy? Was anyone, name one, person forced to believe or swear allegiance to any diety?
By attempting to provide support for every moral view point you provide support to none.
Our rights either come from God or they come from man. If they come from God, as our founding documents adhere, then they are out of man's reach. If they come from man, then men can change them as they will.
The path plotted by you is the latter of these paths, whether you choose to recognise it or not. Men are infinitely fallable, corruptable, and false. By removing God from the public square we have opened the door to wolves. Don't be surprised when the secular moralism you trumpet turns out to be an empty promise, it hasn't the capacity to be anything different.