How is recording the sex of a human at birth harmful to transgender people?

Because it "invalidates" their pretend identities. The NEJM piece takes the position that BCs are meant to provide personal "validation." Which is BS. Anyone who sits around all day fretting over the sex markers on their BCs, passports, drivers licenses and feeling "erased" and that their "existence is being denied" coz of them is, as you say, mentally ill - and it's their mental illness that's causing them harm, not the documents that reflect reality.

This is akin to people over 65 saying that their Medicare cards and senior citizen discount cards and transport passes are "harmful" coz they make us feel old. Can you imagine people with disabilities who have vehicles with license plates or badges showing they can park in handicapped places saying this makes them feel invalidated and singled-out as having handicaps? Or overweight people who need seatbelt extenders on airplanes saying that such devices shame them?