Not to be picky, but...America was not created by a Christian society, nor were they seeking religious freedom. It was created by a small group of secular free thinkers who went to great lengths to separate Church from State in an effort to, in part, prevent the religious violence that had devastated England for the last several centuries.
Some of them were Christian, some were Deists, others were Theistic Rationalists.
'In God We trust' didn't appear on coins until 1864, and not on paper money until 1957.
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But yes, I agree with sentiment that there's no need for hyphenated citizenship, in name or ideology. The Statue of Liberty's "Give me your tired, your poor..." poem ("The New Colossus" by Emma Lazarus) is pretty clear.
But I also recognize that's not necessarily a mainstream opinion these days. And I'll leave it there to avoid slipping into politics.
An 1883 poem on an 1886 statue...since we're listing dates.
The Age of Reason/Enlightenment certainly influenced some of the 'founding fathers' and their outlooks, and I probably should have said Judeo-Christian, which is a little more inclusive.
Deism - the idea of "the existence of a creator god who established the universe and its natural laws, but does not intervene in the world after creation" (Google AI) was definitely influenced by the logic/rational approach at the time.
As always, the pendulum swung back to Romanticism "an artistic, literary, musical, and intellectual movement that originated in Europe toward the end of the 18th century and flourished until the mid-19th century. It was a reaction against the Age of Enlightenment's emphasis on reason and order, valuing instead emotion, imagination, individualism, and the appreciation of nature." (Google AI)