Absolutely as intended. We have their writings that specifically speak to their intent. A relevant Jefferson quote being the actual source of the phrase 'separation of church and state' itself:
"Believing with you that religion is a matter which lies solely between Man & his God, that he owes account to none other for his faith or his worship, that the legitimate powers of government reach actions only, & not opinions, 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." - Thomas Jefferson, Letter to the Danbury Baptists 1802
As you can see he clearly quotes the exact text of the First itself and elaborates on its meaning and goals.
You are not the expert you claim to be and you are wrong. Jefferson, who was the personification of contradiction, did not write the amendment and his separation language that “y’all” love to quote is not indicative of the founders’ intent. It was one statement he made. In fact, Benjamin Rush, who some call the father of free public education in the states, was a proponent of the Bible being taught in schools.
As evidence, Jefferson’s language could have been the language of the amendment and, yet, was not. Nothing like it. His separate writing is not the constitution.
Meanwhile, for the Historian should know, after the amendment was passed, there were states that had official churches and the Bible was taught in public schools and the Ten Commandments were on the walls of government buildings and schools, and “in God we trust” is on our currency, and …
States were clearly not intended to be covered. And, no one thought that having the Ten Commandments posted on the wall was anything like an establishment of religion until secularism became a religion and atheists who claim to not believe in God became obsessed with God.
Go spin your BS someplace else.