Yes, "sex" could just not be on the birth certificate — nor on any other identity document that follows a person around for life and is difficult to change. That might be the best option, given a sexist/transphobic society. Some people have been working on this. According to this 2021 article, "the American Medical Association now recommends removing sex from the public portion of birth certificates." https://www.bu.edu/articles/2021/eliminate-sex-from-birth-certificates/
But trans people shouldn't be required to have inconsistent sex/gender markers. Any discrepancy (one ID with another ID, or one ID with our lives) makes it difficult to do anything for which we need IDs. Having one ID to describe our bodies (which can change in certain important ways) and another to describe our presentation/behavior/identity (which can also change) is just compounding the probability that the little static binary labels won't accurately describe the complexity of our lives. Giving people ID with a distinction between physical sex and social gender amounts to forcing trans people to carry a "trans identity card" in a profoundly transphobic society where the last CPAC speech was literally "eradicate transgenderism". At the very least, until prominent conservatives stop receiving Republican platforms to say "destroy transgenderism," trans people don't want ID cards that essentially point out that we're trans. Most of us will never want an ID card pointing out that we're trans, no matter how polite Republicans are to us. If there are "F" and "M" options, the trans person can be allowed to choose. Maybe the F/M designation originally means "genitals observed at birth" when it's first written down, and a few years later it starts to mean "however the person identifies and lives." It is an alphabet letter and therefore its meaning is flexible. It can even mean different things for different people. It does not always work the way we expect it to work. As long as there is a sex/gender field on the document, most of us want and need that indicator to reflect the sex/gender in which we are actually living, or whatever is going to save us from discrimination and violence. Our F or M hurts no one else. Some places offer an X, and that hurts no one else either. It is our identity document that we use to facilitate our own lives.
A birth certificate is requested in situations where people want information like "Are you a U.S. citizen?" There could be different, better ways to ask for whatever information it is that they're seeking. It is hard to think of situations in which anyone truly needs to know what genitals the nurse saw when we were born, and anyway that information doesn't need to be on the same document we use to prove the city where we were born, which is also the document we use to get other identity documents.
