Awhile back before same-sex marriage, certain states refused to issue amended birth certificates after an adoption in the names of both same-sex parents. In essence the refusal to amend a birth certificate was a state’s tacit way of disapproving of the adoption.
Today that is no longer the case for same-sex adoption and birth certificates as far as I know for gay married couples.
However, the issuance of birth certificates is still a thriving tool in another area of law – children born in the United Staes to parents who are not US Citizens.
In THIS recent NY Times article, a mother who gave birth in the US but was here illegally learned she could not satisfy the Texas requirements for a Texas birth certificate meaning she would be unable to prove her child’s US citizenship.