I had to think about that for a second, I was born in the USA to two US citizens, but that does not show on my birth certificate, so yes I must present proof that not only was I born in the US but that both my parents were US citizens at the time of my birth. Lacking that the USSC will have to decide if one or no USC parents is sufficient to be a natural born citizen. This has been my argument all along, but the case puts it more clearly.
My birth certificate ( and that of my daughter) shows very distinctly what nationality ( ie where born) my parents (and her parents) were. Now I don’t know what info your state required but at least in Colorado and in California that infor is required.