You can do that and that will take care of the "pregnant man" case, but it won't stop a hospital from accidentally recording all births as still births, or other mistakes.

Some things that might seem like mistake aren't mistakes, e.g. people over 18 having pædiatric procedures (e.g. undecended testes is often caught shortly after birth and an orchidopexy is often done in pædiatric wards, but if it was undetected and ignored, for 20 years a 20 year old might need it.).

It's hard to solve a people problem with technical solutions.