Which one would you raise to preeminence over all others as a basis for national morality?
It is also clear that the religion they practised, and used as the moral underpinning, was Christian. In fact, at the time, because of the nature of Christianity, when practised as advertised (love they neighbor as thyself, do unto others, etc), it was probably the only persuasion at the time that could make the morally responsible liberty espoused by the founders work.
But I do not believe they intended it to be defined only for a specific religion. As I said, as long as the foundational moral principles are in place and unalienable rights are recognized and vouchsafed, along with the understanding that they come from God, it will not matter.
Therefore, an understanding that we are "under God" is intrinsic to the understanding of the basis of this Republic.
Basically every monotheist religion shares the same moral guidelines, which, coincidentally enough, were the guidelines upon which our country is founded as well.
So in your world, you don't want any religious values informing law.
What, pray tell, would your standard be?
Your own mind, or someone else's?
Hint: Those are the only two choices left. So in a world which rejects all religion based morality, whoever is the big dog calls the shots. That's the way it ever was, and ever will be.
If you haven't heard, God is a generic term recognizing a supreme power, not the exclusive name of the deity worhipped by Christians. The Bible identifies his name as YHWH in Hebrew, translated by some as Jehovah and others as Yahweh and some others as Yehowah.
Under God simply means recognition of a higher power, whatever you may regard that as.
To deny that, as atheist and human secularists do is tantamount to declaring their religion a state religion. Yes, they are religions as well.