Birth certificates have never been about "registering who the biological parents are". Ever.
If that was the case, you'd need a DNA test to fill in a birth certificate.
Birth certificates exist to register the LEGAL parents.
Under UK law, a married man is the legal father of any child his wife gives birth to, even if there's no way he can be the biological father ie they used donor sperm. (Unless there's a paternity challenge and the court rules a declaration of parentage, which is relatively rare.)
That's why married women can put their husband down on a birth certificate without him being present (and could legally put their husband's name on a birth certificate without his knowledge or consent) but unmarried woman cannot legally fill in the "father's name" part of a birth certificate unless the man in question either attends in person, or submits a Statutory Declaration of Parentage form via post.
If couples use donor sperm, the donor's name would not be on the birth certificate.
Mumsnet is generally not well-educated about the law over birth certificates. Whenever someone starts a thread about being abandoned by their boyfriend after getting pregnant, or starts a thread saying they're pregnant from a ONS and the man doesn't want to know, there are always loads of replies saying "whatever you do, don't put his name on the birth certificate." But a woman can't legally put a man she's not married to on a birth certificate without his signed consent.
This is just another way for the far right invasion of Mumsnet to rant about "wokeness" by pretending to care about children while happily throwing lesbians under a bus. (Feminist chat has always had a problem with homophobia and lesbiphobia.) If you genuinely believed this harms children you would be petitioning to an end to the UK law that holds married men as the legal father regardless of biology. Or you'd be lobbying to make it illegal for married couples who use donor sperm to put themselves down as the parents.
But you won't because those are married heterosexual couples, and not gay people.