Such a document is not required by the US as a condition for the children to travel with one custodial parent to exit or enter the country of citizenship for those children who have already been issued a valid US passport approved by any and all custodial parents.

My extended family has at least a million miles of international travel under the belt where the US children have traveled with only one custodial parent without any such notarized or otherwise easily forged letter in tow from the other custodial parents.

Airlines and foreign governments may require more than the US, but sometimes even where there is no requirement airline and passport control types can make up things and pretend as if such a letter at the airport is a requirement even when it in fact isn’t a legal requirement.

If worried about being denied travel or entry/exit because of this, take a signed letter along with copies of photo ID on it and do a FaceTime/Whatsapp/Signal call with the other parent and let it run from there.