This is classic secular humanism and relativism. The phrase, "separation of church and state" is widely misunderstood and misused.
Our nation was founded on a bedrock of Judeo-Christian values. These values were codified in secular form while expunging its Author from public discourse. Indeed, without this bedrock our Founding Fathers concluded that our country could not endure. The main difference between the bloody and failed French Revolution and ours is that we adopted an immutable set of values from a Higher Power that the Jacobins in France did not.
Nowhere in the Constitution is the phrase, "separation of church and state" used. This first appears in Virginia where the Baptists sought elimination of the official state religion, Anglican (Episcopalian), because they felt their ability to practice their religion was threatened. The Founders widely agreed with the Baptists, and thus encoded this idea in the First Amendment, as you quoted.
However, the Founders, even those who were not Christian, thought religion essential. None would have thought to toss out the very basis of our eduction, laws, and culture (Christian Paideia).
Americans understood that Freedom without morality quickly degenerates into debauchery. Tragically, we are now seen this degeneration and its ugly consequences in real time.
Bill Flax wrote in Forbes magazine, "Whether from sincere faith, or, prudence instilling an honest, law-abiding, responsible and hardworking populace, all esteemed biblical morality as the bedrock of self government. George Washington believed, 'Religion and morality are indispensible supports' for 'it is impossible to rightly govern the world without God and the Bible.'"
The very framework of our Constitution was based on the idea that God clearly instructed us what is sinful.