Your child is still a citizen of the USA by default if you are (even if they were born in another country). This doesn't protect them from the requirement to sign up for selective service for example.
You can surrender citizenship, but then you would need a visa to stay in the USA.
To obtain a passport you (or one of your child's other close relatives) would need to sign an affidavit attesting that they were born when and where they were born. If your child is orphaned and they don't know who delivered them this is going to be an absolute nightmare for them
