Birth certificates aren't medical records. They are the government's "vital records." They're used for administrative/legal purposes.
In the US, when a child is adopted, their birth certificate is commonly amended to reflect their adoptive parents' names instead of their birthparents'. The original birth certificate is not necessarily available to the public nor even to the person whose birth certificate it is. You'd have to seek a court order to see the original. A judge might not grant the request if it isn't your business. Obviously the historical & biological fact of one's birthparents can't change. But City Hall administrators can deem the details of the birth/adoption as socially and legally irrelevant and also a matter of personal privacy so that birthparent names won't follow an adopted child on their birth certificate for the rest of their life.
We show our birth certificates when we get married or divorced, when we get other identity documents (drivers' licenses, passports, etc.), when we apply for government benefits, or when schools or employers ask for them.
If a birth certificate contains information like names, genders, etc. that don't match the way a person lives, it can "out" the person unnecessarily when no one had even intended to look for those particular details. If you're in HR and it's your job to ask for employee's birth certificates as a routine matter, you don't need to get information that allows you to say "oh hey cool you were adopted, your birthmother is my next-door neighbor, and I never would have guessed you were born with those genitals."
That is why transgender people are allowed to change our birth certificates. Most U.S. states have allowed it for decades. We need this for the same reason we want an appropriate gender listed on our driver's licenses. When we're pulled over at a traffic stop, there's no reason for our driver's license to out us as transgender to the cop. When we enroll in a university, there's no reason for our birth certificate to out us as transgender to the staff who put our data in the system.
The birth certificate, like the driver's license, is subject to human error/omission at the time it's drafted, it can be legally changed, and thus it isn't the proper tool with which to communicate biological facts if/when it's ever important to communicate those facts. If a doctor needs to know our sex as it was recorded at birth, or something about our birthparents, etc. we'll tell the doctor that medically relevant information. The doctor doesn't go to City Hall and request a $35 certified copy of our birth certificate to find out how we were physically embodied as infants.
