The term "the church" is all inclusive and doesn't consider the dramatic differenses from one denomination to another. The question you have formed here has overlapping apparent facts but separating them out might help. I find the following helpful in understanding most issues regarding the Separation of Church and State concept.
First, the First Amendment of The Bill of Rights does not have the words "separation of Church and State" anywhere in it. The concept is there but it is one sided, at best. The one side it rests on is the side that has the government keeping out of the "doings" of the Church. Also, the government is not to sanction any one religion as "The Official Religion of the Country.
Secondly, even if one could pass a law that would impose "a separation of Church and State", it would be impossible to truly achieve. True Religion takes place in the heart of a person...at the core of ones being. From here is where the character of a person grows and is what the world sees. One must realize that the character of any one individual is formed in such a manner that the character is as much a part of the person as the color of ones hair. You can change the outside color but the true color is on the inside and, with time, the true color will show through in the end.
In conclusion, those that we place in office as President and Congressmen are all people with characters that were formed over the many years of their lives. If they are religious in any way, that religious aspect of them can never be separated from them and will influence their decisions throughout their tenure in office.
Let me just say that the level of commitment that any one individual has regarding their religion varies. Thus, one will see people that could be cosidered hypocrites attending Church. It is my experience and understanding that the Bible speaks of situations where there is no separation of church and state. Look at King Saul and David. Even before Israel had a King, the leadership were the spiritual leaders of the tribe of Levi, as were Moses and Arron. Even Jesus, when the soldier came to Him to heal his daughter, Jesus didn't scold him for being a soldier. The topic never even came up.
I hope this helps!