The only person, after childhood, who needs the birth certificate is the person whose birth it records. The information should be as accurate as possible.
You're right it's the child's.
To have a lie on the birth certificate had been devastating for people I know who have discovered the truth later in their life.
It's not just a piece is paper representing who brought you up as mum and dad, mum and mum or dad and dad. It's your lineage. It's where you got your eye colour, your height, your sticking out ears, your rare genetic blood disease, your badly functioning kidneys from.
It's the piece of paper that stops you inadvertently marrying your half brother.
And it doesn't give a stuff about the sexual preferences of the people who brought the Child up and wether they are currently treated fairly in law and in society today. It's not about them