If the child was seen by the parents (both parents if it’s a married couple) before the adoption takes place in the foreign country, then it is necessary to readopt. In some countries the adoption takes place before the parent sees the child. In some countries (like Korea), the child comes here still a citizen of that country and the adoption takes place here. If only one parent of a married couple travels to the country where the adoption takes place, the adoption is not official here when the child enters the country. In that case the adoption is not official until the state of residence recognizes it, which often requires readoption. And the child is not a citizen until that takes place.
When I adopted in 1996, it was necessary for the parent(s) to apply for a certificate of citizenship for the child to become a citizen. After the Child Citizenship Act, citizenship became automatic when the conditions in paragraph 1 (lol) took affect but the parent(s) still had to apply for the certificate. Now the certificate is sent automatically a few weeks after the family enters the U.S.