Issue 1 - A karyotype that is as accurate as the age-old glance at the newborn's genitals would require lab work. Mandating such a change at scale would see push-back from almost all of the parties involved, from new mothers to insurance providers.
Issue 2 - You are suggesting an objective and emotionally (fairly) neutral fix for a hot-button culture war issue. While admirable, this is much like hoping that pacifists will win a shooting war.
For what purpose? It's not a medical document anyway.
My guess is same argument as adding blood type to ID: it could help you in situations like emergency or accident (checking blood type is a few seconds anyway, but there might be situation where a test kit is not available when needed).