I could retort (being faithfully “contrarian”) that if we ever leaned on “government” or even “society” to guide us as to what was Christian, we went to the wrong places. Even in a time that saw more friendliness to the faith. But it’s true that “Because of iniquity, the love of many will grow cold.” All some people knew of the Lord was indirect, through society or government. And when secular trends began to fight against that, those people fell.
I’d like to beg the indulgence to assert that we should welcome secular friends to the faith, but never ever lean on them. Let a stout private fire to the Lord burn in our hearts, one that does not depend on secular fuels.
This issue is one among many that I often contemplate. As someone I used to know said: "The law is a normative force."
There is a huge segment of "the herd" minded people who will go along with things because they are officially sanctioned. The fallacy of popularity is only a logical fallacy. Emotionally it is a very valid principle. Most people want to fit in, and will follow whatever ideas are considered proper by the majority.
Women are especially conformist to the the herd mindset.
Your point is whether or not there should be any compulsory Christian (or other religious) doctrine in governance. I can't really say if this is correct or not beyond noting that when there was, the USA was a lot more moral and behaved a lot more responsibly.