There is of course another problem with birth certificates (at least historically)
When I was trying to track down a possible relative who might have been "adopted out", I was looking for a birth registration under a number of possible names (and having found it and no further records for that name I was planning to buy a copy of the certificate to see if it was marked "adopted").
This led to an interesting email correspondence with one of my local registrars about "historic practices". In essence (at least in the 1950s) a woman could walk in to a registry office, claim to be Jane Doe wishing to register the birth of Joe Doe, claim to be married and give her maiden name as Smith - and a Birth Certificate would be issued without further questions or requests for evidence of proof of identity, marriage or even of the child's existence.
What is the current practice for proof during registration?