he people making that history were strongly influenced by the Bible
Depends what era you’re talking about and what you mean by influence. I would say that the reason Thomas Paine’s Age of Reason was so popular and that Jefferson made his own version of the New Testament, which removed the supernatural, suggest that the Bible was less of an influence in the founding of the nation than would be supposed here. The fact that Muhammad is in as venerated a place on the Supreme Court building as Moses also suggests they didn’t think it was the source of all wisdom.
Really, you need to look no further than our legal system though to see how little influence the Bible and Christianity actually have. I don’t just mean the First Amendment, I mean the fact that our whole system is basically a gradual evolution from the laws of Ancient Rome. They had trial by jury in Ancient Rome. It was a permanent jury, not a jury of one’s peers, but you can see the skeleton of our legal system and how it came from those ancient heathens, not Jesus.