When government controls a religion and has a state religion history shows that it is bad. Because wicked people eventually take over governments, even if only for a while, it is better to have a separation of state and church. Adolph Hitler and Germany of the Reich era had an official church and people like Karl Barth left it, or wouldn’t join it. Not even Christians agree on correct doctrine not uncommonly, imagine if President Trump, Hillary Clinton, Elizabeth Warren or Bernie Sanders made the theological decisions as leader of the Church of America.
State churches aren’t good for either Christians or secular citizens. I would say the Church of England is somewhat poor on theology and discipline. England led the way on homosexual marriage and the Anglican Church of Canada put a heavy price on native Canadian children. With a state church theological pursuit of truth is limited to what the state wants. The U.K. has religious pluralism though it has the Church of England. There are still Catholics and other protestants. I am not a fan of royalty or authoritarianism; some people are.
The Roman Empire divided with the state church of the Eastern Empire at Constantinople. The schism led to the eventual sacking of Constantinople by crusaders and the later conquest by Turks and more war. I can easily imagine a different course of development for Christian history under different circumstances; the secular and political condition shouldn’t really decide how Christians form ecclesiastically. I like a priesthood of believers church structure rather than a patriarchal structure.
If government is able to influence religion that works badly for the church and freedom in general-even in the secular world. Sin in the world in Europe and North America would emerge as a certain percent of the population regardless of the relationship of government to the faith community. It was the government-the-US Supreme Court, that forced sin on the nation (homo marriage), reflecting elite desire for sin and populist sales.