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Being able to prove citizenship at a traffic stop or while walking down the street on an ordinary day would be much more difficult than being able to prove citizenship at the pleasure of investigators.
Being able to prove citizenship when someone committing fraud has stolen your identity could be downright troublesome.
Being able to prove citizenship without the cooperation of a government agency, some of whom may not have a motive to help you, might be practically impossible.
Government documents are key items of evidence in citizenship claims. Without them, we can really only rely on witnesses to a birth and raising to adulthood of a child. Those same governments might well be the agencies actively prosecuting someone whose citizenship is in question.
Assuming that we can all easily prove citizenship, on our own, when others challenge it shows just how little we've thought through this process. Compare the concept of proving citizenship to the idea of proving other observable facts about a birth besides location. Hundreds of millions of American citizens throughout history have had no reasonable, provable information about the specific circumstances of their birth, throughout history.
I think the question is so simplistic as to lead us to some unrealistic conclusions about our thoughts on the security and reliability of proving citizenship.