America's always been a "credo" nation, and has always been pretty explicit about what its credo is. "We hold these truths to be self evident; that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty, and the pursuit of Happiness." "I pledge allegiance to my Flag and the Republic for which it stands, one nation, indivisible, with liberty and justice for all."
In other words: freedom, equality, justice, democracy, and self-determination.
America's relationship with religion has always been complex. The first amendment simultaneously guarantees the free practice of religion and enshrines its separation from the state; as a government, the U.S. cannot make any law that preferences a particular religion. Many of the founders were Deists, which is a religion that simultaneously accepts the existence of God but also relies upon empiricism, rationality, and individual thought to divine His will. America is also explicitly not Protestant, and the dominant religion varies a lot from place to place. I grew up in New England, which has a Puritan early history but now is populated by largely Catholic Irish & Italian immigrants. My mom's childhood friends in NYC all seemed to be Jewish. In Utah, it's mostly Mormon.
I'd argue that the main reason America is falling apart is that we've forgotten those core values of liberty, equality, justice, democracy, and self-determination. These have always existed in some tension, but right now we're not doing a particularly good job of any of them. Instead the focus is on capitalism, virtue-signalling, protection from outsiders, preferencing your in-group, forcing your particular morality on others, and many other very human but very un-American activities.