This is such an absolutely bizarre and downright counterfactual idea to enshrine into law.
Birth certificates are records documenting what actually, factually happened at a particular place and time.
When Christopher Hitchens was naturalized as an American citizen (
back in 2007) we didn't falsify his original place of birth (scratching out Portsmouth and penning in Washington D.C.), we just gave him all new paperwork.
When I was legally adopted in 1994, we didn't change my name on my birth certificate (a document from two decades earlier) but instead I was issued a brand new set of papers which would come in handy whenever I needed to prove that the person in the adoption paperwork is the same person as the one on the birth certificate.
Why would activists for any cause insist upon the right to falsify historical documents?