“There is a broader, similar question about citizenship in general. Is any citizenship automatic, or does all citizenship depend on showing suitable evidence (thinking of the person born in the US to US citizen parents, but without a BC, for example) to the government.”
Citizenship. The status of being a citizen (Black, 6th ed).
Citizen. One who, under the Constitution and laws of the United States, or of a particular state, is a member of the political community, owing allegiance and being entitled to the enjoyment of full civil rights (Black, 6th ed).
Vest. To give an immediate, fixed right of present or future enjoyment.... To accrue to; to be fixed; to take effect (Black, 6th ed).
Note, citizenship grants civil rights and the protections of the political community in exchange for the duty of allegiance and subjection to the dominion of that political community. While a foreign born person may be offered the civil rights of citizenship, the foreign born person may not be obliged to accept the offer of those rights and duties associated with citizenship. The vesting of the civil rights provided by the membership in the political community represented by citizenship may be contingent upon the person performing the duties required to vest those civil rights to the satisfaction of the political community. In other words, although the civil rights associated with citizenship may be granted at birth, they may not become vested civil rights until and unless the duties required to vest those civil rights have been accomplished.
I say that citizenship is automatically vested by operation of 8 USC 1401(g). 1401(g) describes circumstances of birth, and does not make citizenship contingent on application for recognition.