Most if not all states routinely issue birth certificates showing the names of adoptive parents, instead of biological parents. And all states routinely issue birth certificates listing whatever man the mother happens to name as "father", with research showing that a large percentage of these are not the actual biological fathers. Birth certificates, at least for the past century or so, have never been intended as a means of identifying a child's biological parentage, but rather as a means of identifying a child's legal parentage -- who has rights to make decisions for the child.
It was my understanding that there is an original birth certificate out there that shows the biological parents. That information may be sealed but it exists. My DH had a son given up for adoption. There are records out there that show he was the bioligical parent, because they had to go to court to adopt the child. The adoptive parents were not official at birth.