If indeed the real world works this way, then that defective administrative procedure is the one to change. Not going around rewriting birth records as a workaround. I have no certain knowledge, but I’d bet you’re wrong in a lot of cases. Though maybe not locally wherever you may be.
IMO Sir T-Cups has the right philosophy. The rest is implementation details.
Here’s a comparison: Real property records in civilized countries stretch back decades or centuries. Recording each twist and turn in how every square foot of land in a country has been divided, sub-divided, re-agglomerated, bought, sold, inherited, and in many cases, leased. Only the most recent record is controlling for current legal reality. But all the history is still there and available for whatever value it may have in the future.
A birth certificate is just that. Evidence of facts true (net of error) at the time of birth. Let’s set aside sex change for a minute and concentrate on something more mundane: name changes.
When Suzy Smith marries Johnny Jones and chooses to take the name Suzy Jones, do we go back and alter her birth certificate? No. Of course not. Instead a legal document is prepared that amends her name. And now her “permanent records” have two documents. And on the strength of that she gets a new driver’s license, credit cards, etc.
When Theodoreus Theanopolopolis decides he’s tired of spelling his name for clerks and changes it to “Adam Ant” do we go back and alter his birth certificate? No. Of course not. Instead a legal document is prepared that amends his name. And now his “permanent records” have two documents. And on the strength of that he gets a new driver’s license, credit cards, etc.
This bit of recognizing that sex/gender/whatever word we want to use is both A) not 100% binary and B) not 100% fixed for life is all pretty new to many people. And is also pretty new to a lot of bureaucracies. Lots of people and no few governments are sticking their heads in the sand real hard over this.
But in principle the answer is easy. We already know exactly what to do because we already do it for hundreds of other attributes of people that we’ve always understood to be not 100% fixed for life. We keep the original records as they were originally and issue a separate amending document of some sort.
I find the accusations of bad faith gender/sex/whatever Luddism in this thread to be overwrought. YMMV.