These are basically my thoughts as well.
I recall a project in the 70s.
A succession of females were asked to participate in some research on their college courses or something of that kind.. They were shown into a waiting room and asked to wait as they were busy.
There was a woman in there with a baby, over a year old. Also, a table with a toy car and a doll. The baby was alternately dressed as a boy or a girl.
Anyway, the woman suddenly said she needed to use the toilet and would they mind holding her baby for a few minutes. While she was gone, if the baby was dressed as a boy, the participants held it losely and offered it the car. If dressed as a girl, they held it more securely and offered the doll.
I recall, when I worked in maternity, the very first thing mothers did was check the sex of their baby.
Some things babys do and need are clearly innate, but I strongly suspect than many are learnt.
I often wonder, if our mother's could answer honestly, would they all say that, before we were brn, were they wanting a specific sex, and we we the sex they wanted?
Sadly, or fortunately, most mothers, I suspect, wouldn't give an entirely honest answer.