First off, I want to be clear that I personally am 100% in favor of separation of church and state.
But, in view of history, I am dubious to what extent such a thing might ever be accomplished, at least so far as the USA is concerned.
This is because historically, religion and the state have usually been the same entity. The ancient kings and queens either were considered to be gods themselves, or the offspring of gods, or otherwise appointed specifically by the god(s). Now, this is not 100% universally true of all ancient societies...but even with a society such as ancient Rome, which had a pretty advanced form of political organization, the politicians had to appease numerous religious factions, and Julius Cesar himself promoted a legendary biography in which his birth story has certain trappings of mythology.
The mythology of medieval Europe included a concept called 'The Great Chain of Being'...according to the Great Chain of Being, all creatures were organized by god into a hierarchy, with humans being above beasts, and with kings/queens being above other mortals but still of course below god and his angels. European kings had legitimacy because their lineage had been made royal by the actions of god.
Ancient Judaism is a mythology most likely constructed so as to give unity to an amalgam of various ethnic groups and to give legitimacy to that group's kings and generals.
If the Jewish prophet Jesus actually lived and was actually executed by the Romans, it was most likely because they considered him a political threat. The term messiah had a political meaning in the 1st century AD more than it did a religious meaning.
The creation of the USA's government by via a written, legalistic Constitution in 1787 is frequently considered to be a milestone in political evolution, a step away from the church/state hybrid. And yet, by 1848, when construction of the Washington Monument began, you can see that historical figures such as an insurgent general had been named as 'Founding Fathers' and were taking on mythological characteristics. The Washington Monument itself is based in part on the Egyptian pyramid, a religious structure.
When Abe Lincoln was assassinated in 1865, there were immediate comparisons between Mr. Lincoln and the murdered demigod Jesus.
By the early 20th century, the legal concept of Sedition had been expanded so as to make certain political speech equivalent to heresy, and punishable to the extreme. The biggest heresy was Communism/Socialism, and well into the 21st century, to be openly Socialist is be effectively ghettoized regarding ones access to governmental affairs and public speech.
US history, as taught to primary school students, is essentially mythology, a story of virtuous peoples fleeing religious persecution, just as the legendary Hebrews allegedly fled the Egyptians.
The US Constitution, considered by most of its writers as a pragmatic piece of political experimentation, has achieved a status of quasi-religious reverence. It is even kept enclosed in glass, just as religious relics are. Arguments regarding interpretation or amendment of the Constitution frequently hinge upon a notion of 'original intent,' which implies that the writers of the Constitution were specially decreed or empowered with political wisdom inaccessible to modern humans.
In particular, a form of extreme fetishism surrounds the 2nd Amendment, wherein a meaning ascribed to it by modern readers takes precedent above every other article and amendment of the whole document, including the preamble.
(An episode of the original Star Trek TV series, 'The Omega Glory,' satirizes a group of neo-barbarians who treat the Constitution as a Holy Relic.)
I think some of the shift of the USA toward a nation based on mythology is probably the result of intentional manipulation, but I think some of it is also unconscious, unintended...the result of primitive mental programs which seek strong and bellicose tribal identity.
I know I took a long way to get here, but the point is that the political and religious regions of our brains are intertwined.
Those of us who may think they are above the primitivism of religion, may themselves fall victim to the primitivism of nationalism or jingoism. Even those who think they have moved their minds outside the reach of the old gods, may yet be intoxicated by the violent ecstasies of patriotic mythology.