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Citizenship actually depends on whether or not the parents were married, also.

If married at the time of birth, and one parent was a US citizen, and that parent had lived in the United States for a period of one continuous year prior to the birth of the child, then the child is a US citizen.

If unmarried, the child takes the citizenship of the birth mother. If the mother is a US citizen, and lived in the United States for at least one continuous year prior to the birth, the child is also a US citizen. If the mother is not, and the father is, then the child takes citizenship of the mother unless:

1. Paternity is proven beyond a reasonable doubt, and
2. The father agrees (unless deceased) in writing to provide financial support for the child until the child's 18th birthday, and
3. The father is a US citizen at the time of birth, and
4. While the person is under the age of 18 years—
(A) the person is legitimated under the law of the person’s residence or domicile,
(B) the father acknowledges paternity of the person in writing under oath, or
(C) the paternity of the person is established by adjudication of a competent court.

8 USC § 1409 - Children born out of wedlock

Because of this section of law, there have been instances where US citizens have fathered children overseas, brought them back to the US without their mother, raised them, only to have the children deemed to be deportable non-citizens in their 20s. Because of the provision in Paragraph 4, if the father never established paternity prior to the child's 18th birthday, the father may never legally establish paternity after the child's 18th birthday.

I just did my first renewal and had to send a copy of my Certificate of Citizenship, even though I did that when I originally got my CHL.

I thought they would assume since I already had a CHL then I originally met the requirements and wouldn't have to prove them all over again.

But, they wanted another copy of my DD-214, also. So I guess you have to verify citizenship and veteran status each renewal.

As for the OP's question, wouldn't a passport be proof of citizenship? After all, you can't receive a passport unless you're a citizen.