A few things.
The first is that the U.S. Constitution was created to limit government power. It says what the government cannot do, not what it can do. The First Amendment was made to protect the People from tyranny.
A little known fact about the First Amendment was that its inspiration came from a desire to prevent the government from influencing the Church, not the other way around. It was believed that when government began to dictate the morals of God then there would be real tyrannical problems like what happened with the Church of England. So... the irony here is that the separation of Church and State is actually about protecting God worship from government involvement. In modern times though, the opposite need has arisen... to protect the masses from the tyranny of the Churches. It functions both ways.
Lastly, it's true that there was a lot of Christianity in early America which formed traditions in government institutions, but that was merely a product of the dominant culture and demographic. Doesn't mean that it's a Christian nation, or that those institutions can't now be challenged under the First. The truth is that government rituals around religion have always been a First Amendment violation, but because the vast majority were Christian at the time, they didn't care.