But there's also the issue of if something happens to a person that requires the knowledge of their sex, and they aren't comfortable disclosing their sex (at least not aloud), such as they need to get an operation, then them having no sex on their birth certificate means there's no way for the doctors to find out their sex (short of forcefully undressing the patient, which is illegal), which can be vital information depending on the situation.
I think "weird looks" (which are highly down to interpretation, people with anxiety for example might assume everyone is giving them weird looks) are too small a thing to worry about, especially when a child is born. You don't know if your child will becomes trans down the line, so you shouldn't assume they will, that's just as bad as denying them the right to choose their gender identity. Their sex has nothing to do with their gender identity so you should just say it like it is, hiding it on their birth certificate doesn't actually do anything for them, certainly not immediately.