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.
It might have, and yet do we want to rejoice in whitewash.
If an eternal score could be kept, one might question whether earthly good manners bespeak the currency of heaven, which is a soul bought by the Son. People can have sub-salvational walks before the Lord, where they are trying to look good enough to the Lord to earn a pass into heaven. This is a common parasitical philosophy in what we might call Christendom. Civic manners will improve; but souls furthered on their way to eternal heaven may not.
God might have good reasons to let the bottom fall out of a civic scene, when the cost of NOT doing so would be the cost of obscuring eternal truth.