Trans people don't change gender. They stop hiding it.
Updating a birth certificate is fixing an error
It's also a strange point to hold ideologically. Why is "at birth" an ideal to hold above others? In literally any circumstance where a birth certificate is needed, "now" is going to be more useful than information that is decades out of date. Nothing is gained from holding to an ideal that puts out of date information above current information, so appeals to treat it as sacrosanct always make me wonder exactly what it is that makes people put pointless ideals above the very real impact incorrect information has