God has chosen His nation already so there's little use in trying to steal that mantle. But interestingly, John Winthrop and the early Puritans did see America as the New Israel. And, AND, no less than Thomas Jefferson and Benjamin Franklin proposed Promised Land imagery for our first U.S. Seal. Call it Providence, Manifest Destiny or American Exceptionalism, but our country was formed with an eye towards the heavens.
In the context of a literary allusion, resistance to tyrants is obedience to God, it also wasn't accepted for a reason.
The Declaration, Constitution, Bill of Rights and those quotes I listed earlier, while not derived from scripture, really do presuppose a decent and moral citizenry, which at that at that time and continuing until the present has largely in this country been supplied by Judeo-Christian teachings. That's the self-governing part. And of course the Founders appealed to an authority above the state as the source of unalienable rights.
An authority above the state is natural law, not a theistic God, and Jefferson explicitly made the point that the origin of rights was in a pre-Christian environment. The idea that Christianity is by definition, moral, is an unfounded statement and the idea that practicing Christians supply are more decent and moral than other people is both false. Deists, agnostics and atheists have all made tremendous contributions to America, and a lot of good has been justified on principled philosophical, rather than religious, grounds.
As for the "tip of the hat." Two things. Having a civil religion with occasional Christian language or references is not in any way the same as establishing a national Christian religion. Would we really be better off as a nation if the great speeches of Lincoln, Washington, Reagan, FDR and MLK were stripped of their biblical touchstones? Should Barack Obama cut out that "I am my brothers keeper" rhetoric? Is nothing gained when a president ends a speech with "God Bless America" or begins his term with "So help me, God"?
I have no problem with biblical allusions, or teaching the bible in public schools with a secular context, the stories it contains have had an important influence on Western culture, but this is different than governing as a Christian nation and supporting particular sects, for instance giving public funds to the Congregationalists but not the Satanists.
Two. I have no idea how often God intervenes into the affairs of man or nations. But don't you want leaders that ask for guidance from this great source of wisdom we believe in? Shouldn't our country have a day of Thanksgiving and acknowledge our many blessings? No one believes public faith should replace private faith but shouldn't we encourage policies and language that fosters the second?
Your rhetorical question about "leaders that ask for guidance from this great source of wisdom we believe in" is problematic, it assumes that there is a great source of wisdom, I would be worried if a politician was to pray for policy and heard a voice in his head, I don't think that it is wise to confuse contemplation with revealed truth, and the assumption that it is good.
The role of the American government isn't to foster private faith, only to preserve an unrestricted marketplace of ideas, your rhetoric is geared towards your ilk, mainstream Christians who feel that religion improves their lives and makes them better people, but that isn't a universal opinion and dissenters shouldn't have it imposed on them, even with a smile.
Freedom and religion flourish in this country because we draw from both but never rely wholly on either.
What other country can you say that about?
You are gauging two separate things, liberty and religiosity, I will say that there are more free countries that have low rates of religious belief, and many autocratic nations with very high levels of religious belief.
I think that genuine freedom starts when one faces reality and begins to look at religious institutions as man-made, it is a good starting point to honestly think about what is right and wrong, without deferring to scripture (even if there is a God and he sent his one son to be slaughtered on behalf of people who never asked for a scapegoat, it says nothing about the inherent goodness or evil of the deity, you can only charge atheists with a lack objective moral justification because of their honesty, God is only good to a higher moral standard, which diminishes his glory or the word good becomes meaningless).
We can freely choose to do the right thing, without fear or hope for reward, and that choice can be more justified than doing it in the name of God. We don't loose our capacity for empathy and compassion when we abandon supernaturalism, we can retain moral principles (such as reciprocal altruism) without religion, those moral instincts are evolved characters, ones which we can trace in other animals. Now giving our behavior objective justification demands philosophy, and asking if there is in fact objective morality.