That's exactly the New Testament idea. Christians are supposed to pray for our leaders, respect them, submit to them (there would be exceptions based on belief and all, but submit), and pray for a peaceful government so we can go about living out our faith.
Someway, somehow, Christians and everybody else actually have to live together. That's the trick that people aren't seeing, for the most part.
I can't think of a single instance where a government-run church was good for the church. We just get exploited. Give us the government, and we really don't know how to live with a different worldview, even though it should be easy for us, theologically.
It's a mess, but it shouldn't be.