You're assigned male or female at the moment of conception.

It's more accurate to say, human sex is determined at the moment of conception. Sex isn't something that nature allocates or designates the way use of the word "assign" here suggests. Best to dispense with the idea that sex is assigned altogether.

You are observed by a doctor as male or female at birth.

Not all births occur in medical settings. The vast majority of humans born over the course of history were not born in the presence of a doctor. Even in advanced medical settings today, the HCP present at birth who checks newborns' health and records essential facts like weight, height, sex and APGAR score often will be a midwife or a neonatal nurse, not a doctor.

Equally important, due to the widespread use of prenatal scanning - long a standard part of prenatal medical care throughout the world - and the increasingly common use of genetic testing especially the non-invasive test of maternal blood known as the NIPT, the sex of a majority of the world's babies is known long before birth. Decades before the NIPT became available, many women had fetal genetic testing through amnio or CVS.

Moreover, even when birth occurs in a medical setting, it's not true that when a woman gives birth the midwife or doctor call out the child's sex like Hollywood movies and TV shows suggest. The HCPs attending a birth often hand the mother her baby and let her see her child first. Or, since mirrors have long been used to allow women to see their children enter the world, many women first see their newborns' bodies - including the genitals - at the same time that everyone else in the room does. Even women who have C sections are customarily given the option of having a mirror situated overhead so they can see their babies coming out of their wombs.

Whatever the setting, and however birth occurs, observation of a newborn's sex is something all the adults present at a birth usually do, primary among them the mother.

Also, depending on the jurisdiction, parents - usually mothers, but in some places both parents, or fathers alone - are the ones responsible for registering an infant's birth with government authorities, or for signing and verifying as true the paperwork that hospitals or other HCPs/HCFs submit to government birth registries that will ultimately result in the issuance of a birth certificate. Legally, the word of an infant's parent(s) is as important - or more important - as a midwife's or doctor's word in determining whether a child is officially recorded as male or female in government documents and databases.

Shortly after birth, HCPs customarily examine, measure and do quick tests to determine whether a child seems healthy and normal. This will involve observing whether the genitals appear typical and properly developed. But the presence of male or female genitalia is something everyone attending a birth sees with their own eyes.

The claims that sex is only observed and known for certain at birth, and that observing the sex of newborns is only something doctors do overlook the realities of what women go through in pregnancy, which for many decades now has involved routine scanning. Such claims, in fact, render the women who carry pregnancies and give birth entirely invisible in the process, or at the very least sideline us and make us seem dependent on "a doctor" (a job title most people still equate with men) to tell us what's what. Even exhausted after having just gone through labor and the pain & effort of giving birth, or just having had a CS, women are fully capable of using our own eyes to see the sex of our babies, and don't need someone with a medical degree to tell us whether our newborn is male or female.

The only exception would be when a woman is given drugs to make her unconscious or out of her mind when giving birth - practices that occurred only in some medical settings in some countries decades ago, and haven't been in use for many years.