You are conflating religious influence on the state with the state's influence on religion. Once Caesaropapism ended, religions could go back to their proper role. Religion was still influencing government in the 1700s and the 1800s for the better (Colonialism was a matter of material concerns, and would have happened, atheist or not [for proof, see USSR],) as government had taken a passive role in religion. Compare the experience of the French Revolution vs the American Revolution. The atheist one was much bloodier. Freedom of religion does not protect the government from religion.
And I return again to Communism if we're going to keep up this consequences argument. Show me one communist (as in, professing communism, I'm not interested in No True Marxist fallaces) government, exactly ONE, which did not quickly devolve into mass murder and wide scale universal repression. Communism is worse, yet I don't hear you decrying it or wanting to have a separation of Communism-State.
Now, I'm going to make a controversial statement, and the only reason I give fair warning is that I want the rest of my points answered. So just because I am going to make a point many will disagree with, doesn't mean I want the rest to get ignored, since evidently that's the trend.
Religion is a part of the human spirit. It's biological. If you quash religion in its benign forms, which let's face it, most religions are very benign, it will spring up somewhere else. It will spring up in the Church of Science (different from real science, these are the "Toxins-Juice Cleanse-Gluten Free-MSG causes cancer-Vaccines cause autism" idiots), where the Bible is replaced by "studies" they read in tabloids. Or it will spring up as mentioned earlier in the form of Communism or a similar system, itself every bit a religion. Mother-Earth environmentalism is a possibility to. You simply cannot crush the human belief in something he cannot prove. Take Mr. Atheist himself, Richard Dawkins.
Another controversial point coming, don't ignore the rest.
He believes there is no God, and has no evidence of that. Obviously, that does not prove there is God, but it does show he is being irrational. The only strictly rational position is, "I don't know.", and anything beyond that is faith. Could there be an invisible incorporeal unicorn sitting in front of your screen right know? There could be. I don't know, and neither do you.