You never know when to expect the Spanish Inquisition! The first immigrants came to America fleeing religious persecution. Our founding fathers knew the dangers of entanglement of church and state. They insured it would never happen here in the adoption of the first amendment of the Constitution.
A state religion requires all citizens to become members or lose their citizenship. With over three-thousand religious dominations in America that isn't feasible, and if it was, how would you pick which will become the official one.
"Render onto Ceaser's that which is Ceaser's and unto God that which is Gods." A Christian is thus admonished, the Bible requires separation of Church and State. There can be no other interpretation of this warning. It also shows the fallacy of the idea America is a Christian nation. Heaven, sitting at the hand of God, is the real Christian nation, not some land on earth.
The Protestant reformation and the chaos of the break up of the Catholic monopoly show the danger of combining church and state. Millions died in the ensuing wars until the Peace of Augsburg in 1555 ended the conflict and established the doctrine "cuius regio, eius religio" the religion of the ruler will be the religion of his citizens. The treaty only applied to Catholics and Lutherans. Other Protestant sects such as the Anabaptists and Calvinists were expressly excluded.
The combination of church and a state still caused problems especially among the Calvinists. That resulted in the Second Defenestration of Prague and the thirty years war, ending with the Treaty of Westphalia in 1648. All this death and destruction shows the danger when opponents each believe, they have God on their side.
This is the problem with the combination of church and state. Religion forces the state into actions that do not always benefit the state but are necessary because they are demanded by God, or at least God's representatives. No temporal leader can refuse to obey God's orders passed through his representatives in the Church, after all you never ask questions when God's on your side.
Considering the problems caused with the joining of church and state involving only Christians nearly five hundred years ago, imagine the problems it would cause in the modern world with thousands of religions and their offshoots. It would be chaos, with each Church/State Theocracy seeking dominance over all the others. Instead of the millions killed during the Reformation, with today's weaponry the toll would run into the Billions.