Just because they are both political instituitons doesn't mean they cant be disconnected.
It is precisely because of their similar political nature that they CAN AND MUST BE SEPARATE. The sovereign will of the state CANNOT BE SHARED with another political association outside itself. Otherwise, such a situation is what is known in political science as a FAILED STATE. The most direct manifestation of a state's sovereign will is that it commands its armed forces - both civilian and military.
One of the most glaring examples of a failed state is lebanon - where the government does not command the military arm of hezbollah and yet exists within its supposedly 'sovereign' borders. It is this same military arm that commits the entire nation of lebanon to a war with israel while it is accountable to none. Same goes with the palestinian authority and hamas. Herein lies israel's dilemma - it is obligated to wield all means within the disposal of its power and mandate to protect itself, at the same time bound by the international community to appear non-aggressive towards its neighbors.
Diplomacy with failed states is a dangerous precedent. By definition, a state that cannot wield sovereign power within its borders has no right to enter the community of nations.
I didn't elect the church into government.
Yes. You elected a person who, as a citizen, has an inalienable right of thought. The officers of the state have dual roles.