I would hope for a world where nobody feels the need to officially falsify their sex any more, where birth certificates state name, sex registered at birth, place and date of birth, and important adults at time of birth, with each of those adults' actual roles specifically delineated, whether that's provision of whole egg or sperm, maybe mitochondrial donation if that turns out to be an important thing, birth mother/gestational carrier/whatever you'd call the actual person who carried and gave birth to the child, whether they have parental responsibility, and maybe (if you want to stick with the tradition) occupation. It would be less simple than Father and Mother, or Parent 1 and Parent 2, or whatever it is at the moment, but a lot more transparent, especially for complex situations.
Yes, you could end up with 6 "parents" listed — mitochondrial donor, female gamete supplier, male gamete supplier, gestational carrier, parental responsibility holder 1 and parental responsibility holder 2 — but you'd know what's what. And most babies would still have something like Parent 1 (whole egg, birth mum, PR), Parent 2: (sperm, PR).
Those things would be unchangeable, but there would be a space for addenda if the person turned out to have a DSD that meant their sex was incorrectly recorded, or maybe if there was a change to the people with parental responsibility, or a genetic test showed a different father to the one assumed, or someone was adopted, or whatever.
Everybody deserves an accurate, unexpurgated record of the important facts of their origins, birth, and adults associated with them.
If sex and gender identity are two different things, and if we can get to a world where there's no stigma or shame or discrimination associated with being transgender, then there should be no issue with being a transman having a birth certificate accurately showing that that person was identified and registered as a female baby. I'm very unhappy with the idea that a historical legal document can be silently retrospectively altered, and it shouldn't be necessary.
The only consideration that concerns me is the protection of trans people needing to travel to countries where they need to go "stealth" or risk discrimination, but almost nothing besides the birth certificate actually needs to state either sex or gender identity — UK passports don't, IIRC. I guess for working abroad, you would need a birth certificate…? I don't know if I would be happy to endorse a system of falsifying birth certificates specifically to fool foreign countries with discriminatory laws against trans people, though.