I dont know if this is why, other than the house of Lords, that England has been so deterred to make Politics secular. But I strongly suspect its at least part of the reason. I strongly suspect if there had been a hint of religion in the causes of the American civil war, America would be no different right now.
Given the extreme bloodshed of the ACW, had religion factored in, there would be two separate countries now. In the age of steam rail, canal boats, and efficient ships, people wouldn't have just stuck in place and endured.
However, I think you are badly off base in the usual European sense, that of conflating faith and religion in American culture. The worst overlap between religion and government here wasn't after the Revolution, or the ACW, it was during the 17th century when local governments at the behest of English governors required church (CoE, specifically) attendance in the mid-Atlantic colonies such as Virginia. Not surprisingly, the very concept of governmental secularity comes from Virginia (reinforced by the mess in Maryland after the Revolution).
It becomes laughable after awhile. The US has never had a state church, yet most if not all Nordic countries until recently have had a state church. Yet the world has been conditioned to believe in some Hollywood-esque framing of the US as run by churches.
Yeah, I guess you are probably right.
I dont think we believe the churches run the US. But its very notable to us how much your Presidential candidates and Senator's court them. Over here, politicians have never really wore their religion on their sleeve. Tony Blair became a catholic, but only after he left office. Theresa May was the daughter of a Vicar and went to church every sunday. In America, much would be made of this. Over here, its a non issue.
I was just looking this up, and by way of example you can see this neatly encapsulated in this quote from Reagan, speaking at a Latter Day Saints church in Utah in 1982.
“But there are other matters that all of us up here must take up. The matter of prayer in schools. I don't think God should ever have been expelled. There is that balanced budget amendment that we must have. There is tuition tax credits for those parents who are sending their children, perhaps, to a church school or an independent school at the same time they pay the full burden for supporting the public school. And I think they should get some recognition of that fact and some relief for the fact that they are supporting two school systems,” Reagan said. “And there is another problem very close to my heart: that more than a million unborn children every year are being denied the right to life. And I think it is time that we decide that unless and until someone can prove to us that the unborn are not truly living creatures, then we morally should adopt the principle that they are, until it can be proven otherwise.”
For us, that is a distinct link between Politics and Religious thought. For us, this is a deeply strange mix. I dont believe we have ever held politics and religion that closely since the restoration of the Monarchy. Well, unless you count the rejection of James II as religiously and not politically motivated.