I think the time is coming where all children should have both their biological parents and the parents who are to bring them up on their birth certificates.
I don't agree with birth certificates for same sex couples having only the couple's names on them; I don't agree that babies born via a surrogate should have only the names of the commissioning parents on them. In the past it was understandable that children born from donor sperm had the mother's partner's name as "father" but with the knowledge we have now that should also no longer be allowed.
With DNA tracing, hiding reality from children is no longer possible, and with all we know about the psychological harm done by lying to (indeed, not necessarily purposefully lying, but concealing knowledge from) adopted children, it should not longer be legal to lie to any children about their origins.
Birth certs should be a record for the child of his or her birth. The information on it should be accurate. Another cert - like an adoption cert, or the old short form cert - could be issued for children with complex parentage, but in no case should the birth cert - a legal document - contain untruths. Nor should it be changed at a later date to contain untruths.
My children have no access to their birth certs (not in UK), but they have adoptive certs which are factually correct. If and when they get access to their original birth certs it is important that the names of their birth parents are on them.
Sorry, off topic, but I feel very strongly that altering factual records is wrong.