I would not wish to have that as the law of the land. It has ‘ID Fraud’ written all over it.
That said, for candidates running for public office having an eligibility requirement that a birth certificate can settle - yes: there should be a legal authority/mechanism that can perform this verification.
There’s no useful information for ID fraudsters on a birth certificate. Newborns don’t have Social Security numbers, bank account numbers, credit card numbers, etc., etc. And it’s not economical to pay the fees for someone’s name, and a fifty-year-old address.