The separation concept is from a letter from Thomas Jefferson to a church while he was running for president against John Adams. Mr Jefferson was agnostic and mr Adams campaign was saying he was going to interfere in local churches.
the point of the letter is to assure the people that the federal government has zero (ie a wall) power to interfere in the church.
That the establishment clause is specifically to stop the federal government from picking a specific sect (ie Baptists , Anglican ect.) See England and the church of England.
the flow of the concept is one way. The Federal government cant force someone to be Methodist for example to live in the USA.
They also cannot stop one from following a religion Except under some very very narrow parameter's.
Generic God Creator concepts are preexisting and over arching as a back drop of the society.
He wrote, “I contemplate with sovereign reverence that act of the whole American people which declared that their legislature should ‘make no law respecting an establishment of religion, or prohibiting the free exercise thereof,’ thus building a wall of separation between Church & State.”
Like most of history it is bastardized by the media or anyone with a gripe.
Note this is only the federal government.
Some states had a state religion long into the early history of the nation, most went away from it as it hindered movement. Ironically they were in the northeast.
Thomas Jefferson wrote to a letter to a Baptist Church from Danbury, Connecticut, in which he explained his beliefs about federalism and the Establishment Clause.
billofrightsinstitute.org