Everybody convenient ignores the fact that when the Constitution was ratified that most of the STATES still had a de facto STATE religion (or at least a strong preference)... owing to their early settlement roots. Quakers in Pennsylvania, Puritans in New England, etc.
This continued until roughly 1820. The Federal Government itself could not establish a religion... that would have violated the rights of the states to establish their own - as they already had done.
New Hampshire prohibited non-Protestants from serving in either house of the state legislature until 1877.
The Mormons had to got to Utah to get away from religious bigotry. Just saying.