I agree that all forms of documentation need a rethink.
What purpose does altering the sex marker on a birth certificate serve? Humans cannot change sex. Your sex is determined when you were conceived and it is merely recorded when you are born.
What is the purpose of changing a factual piece of information to something that is not true?
As far as I'm aware, in the UK, a person can change their name to anything they wish. As long as it's not something explicitly offensive, you simply change your name via deed poll. In Scotland, I don't think you even need a deed poll.
If your name is Bob and you don't like your name, and prefer to be Sheila - go ahead and change it. There are no laws that say only males should have these names and females have these names. It's merely social convention and stereotypes around sex.
What's wrong or shameful about a man named Sheila? It might be unusual or rare, but nobody is prevented from changing their name.
Human hair in both sexes grows long naturally. All men would have long hair if they didn't cut it. It's the most natural thing in the world. You're a male who doesn't like your hair short and prefers it longer? Go ahead and let it grow. There's no law that says men's hair should be a specific length.
Lots of men also shave or wax their body hair. Again, nothing to stop you from doing the same.
There's no law to prevent men from using makeup or wearing clothes usually worn by women.
You don't need to be 'trans' or claim to be the opposite sex to do any of the above. They are all tied to social conventions and stereotypes associated with the sexes. You won't be struck by lightning if you break conventions and stereotypes. Trust me.
Some narrow minded individuals might laugh - tell them to fuck off. If there's harassment, threats of violence or direct discrimination in employment etc, then we have laws to protect against that.
No male should face discrimination for not looking or behaving like other males. Similarly for females.
As a society we don't have to indulge in the nonsensical fantasy that humans can change sex in order to enable people to style their hair and face as they like or wear what they choose or use a name they prefer.
So what exactly is gained by modifying the factual marker of sex on documentation?