The folks who architected this nation most certainly did, and since they're the ones who devised the sovereignty of the states and embodied the individualism and self-reliance and runs undercurrent to your claim that Americans "hate the idea of being obligated or beholden to anyone else" I think it's perfectly fair to characterize the debate in those terms. And they did in fact see these as universal rights: "We hold these truths to be self-evident, that all men are created equal..."
Anyways, specifically I take issue with your claim that "the idea of "human citizen[ship]" is anathama to American thought, because citizenship implies reciprocal rights and obligations." Obligations, yes, but it does not follow that the extension of our recognized rights to all people is in any way contradictory to our ideals, your personal views on collective rights notwithstanding.