I think it would be a good idea to keep it in the official birth record, but remove it from the printed copy that you get. If someone has a legitimate need to know then they can apply to get the full record, but in normal use it wouldn’t out any trans people.
What legitimate need to know someone’s birth sex is there that can’t be found out more accurately with a physical examination?
Isn’t this already required, under Trump’s executive order? The newborn child wasn’t producing either sort of gametes at conception, therefore it cannot legally be either sex. And since, according to that same executive order, there are only two sexes, the only option is to not list sex at all.
Since birth certificates are issued by state-level agencies, I don’t think an executive order can force them to change how they do it.