When the American founding fathers set up a system in which church and state were kept separate, they had no intention of separating the state from religion. What was at stake at America’s founding was the question of sphere sovereignty. The Constitution separates the mechanisms of the institutional church from the mechanisms of the federal government, allowing both spheres to flourish without interference from the other.
This meant that church ministers didn’t have any authority over statesmen in the federal government, but it also meant that congress couldn’t interfere with the freedoms of the church.
Even this sphere sovereignty originally only applied at the level of federal government and not state government, as Daniel Dreisbach has shown in his book Thomas Jefferson and the Wall of Separation Between Church and State and as I have shown in the following articles: ‘Bill of Rights‘ and ‘Free Speech and Animal Cruelty‘ and ‘The Constitution and States Rights‘ and ‘The Constitutional Convention and States Rights‘.)
These distinctions are often confused.