They get mixed indirectly and very powerfully. There are 2 different issues with the state controlling religion
1) People don’t get to choose how they practice their faith

2) That very same “faith” can then much more easily be used to brainwash, control, and manipulate people for political purposes


#2 is the real reason why church and state are combined. Very few people in power are ignorant enough to believe #1) is the real reason (ie, need to spread God’s word). Except for a few religious leaders, some of whom actually believe it.