The US Constitution is antithetical to European monarchies of that time, especially that of England.
In the centuries before the writing of the US Constitution, Europe was ravaged by wars among Catholic heads of state and Protestant heads of state attempting to impose a state-sanctioned religion upon the people.
If the head of state was Catholic, that state was Catholic. If the head of state was Protestant, that state was Protestant. When Protestants rose to power in Catholic countries, and Catholics rose to power in Protestant countries, that nation would possibly be destabilized.
The Founding Father's wanted to avoid the Old World's strife and instability caused by state-sponsored religion, hence, the Separation of Church and State.
As of now, Fundamentalist Christians are united by their common struggle against the "immorality" of a society ruled by secular Government.
But once they obtained power, how long would it be before they fought among themselves just like the Europeans had?
Which Christian Denomination's or Sect's interpretation of Christianity would be the basis of a non-secular Government?
Would Darwinism be allowed to be taught?
Would Saturday or Sunday be the Sabbath?
Would Christmas trees be banned because they have their origins in Paganism?
Would Congressmen's and Senator's mistresses be burned or stoned to death for being harlots?
Things would get messy quickly.
The Founding Fathers were capitalist merchants who did not want religion's intrusion into government to destabilize society, Government, and the economy.