Interestingly, one of the running themes in the OT was the discontent of the Israelites with theocratic governance. They clamored for a more tangible earthly government in the form of a human king, which God allowed as a sort of “be careful what you wish for” learning experience.
The Christian-era version of the same impulse has been to throw support behind earthly governments to whatever extent those governments make some show of embracing Christian values. If that “nod to God” is there, that seems to be enough to sanitize the worldly aspect and make the government appear to have God’s smile.
Maybe it just comes down to the desire to cover all one’s bases. It’s comforting on one level to think that God is working behind the scenes in your best interests, but it’s also nice to think you have the easily tangible power of the military and law enforcement on your side.