You have that backwards.
They did not want the government to have a say in church matters. Which was indeed the case in Europe.
Religion in Europe is a "private matter" in that you can believe as you like as long as you do not let it influence your actions, teach it to your children or tell anyone one about it or practice any of the rituals anywhere but in your own house with the shades drawn.
In the US the entire thing was turned around. You were allowed, even encouraged to bring your religion out and parade it around the public square.
People could accept or reject it as they wished but you need not keep it secret.
“They did not want the government to have a say in church matters. Which was indeed the case in Europe.”
I disagree. Governments in Europe in the 17th and 18th centuries certainly had a say in church matters for those of the wrong sect.
Was religion in Europe “a private matter” for the English Pilgrims, the Huguenots, Puritans, Quakers, English Catholics, the Amish, Radical Pieists, etc.? They all fled Europe to America to escape church-sanctioned, government persecution and civil war in Europe.
Religion is an individual’s chosen path to salvation. Government is simply a necessary evil, not a path for salvation.
Religion comes first and is not to be subjected to the will of government just as one’s soul is not be subject to the will of the world.