That would be the 4th Amendment[1].
I really don't think that Americans have never cared about privacy; I think if that were true, we would have a different 4th Amendment. Instead, I think that the various amendments and overall content of the Constitution serve to limit the power of government, and in that regard, it behooves the individuals within the government to disregard as much of the Constitution as they can get away with, thereby acquiring more power for themselves.
That the American people have allowed the individuals within government to disregard the Constitution so flagrantly, to me this indicates a deep gap between the ideals of our predecessors and the ideals of the current generation. So, it's not that Americans have never cared about privacy, it's just that the people of today don't seem to have the political willpower to get it back.
[1] The right of the people to be secure in their persons, houses, papers, and effects,[a] against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.
I'm not sure how the NSA thing is justified, especially given the executive order mentioned in the article.
True, many of them may not appreciate the degree to which their lack of privacy is exploited for commercial purposes , but make too much of a fuss about that and people think you're a proto-communist or something. I'm very privacy-oriented by temperament but I've gotten used to being out of step with almost everyone I know over this.