Last week I got to meet my cousin’s daughter for the first time. It was a family dinner and after everyone had eaten she pulled out her barbie dolls to play with. Since I‘ve been on HN for a few years now and this is not the first time I came across this topic I asked my cousin: Why does your daughter play with barbies? Did she choose the barbie doll or where they given to her?
My cousin, confused at first (because remember people: Most people don‘t really care that much about this stuff), answered: I think she chose herself, or maybe grandma gave one to her first.
So this could be it: The little girl might be free to choose as far as her mom is concerned, but is she _actually_ free to choose because she got influenced by her grandma first? Who‘s to say?
I hated them. They were obnoxious, lacked creativity, and smelled weird. I was a very willful child and loud about it, so I obviously refused to use these gifts in any capacity, but I think my refusal to play with toys I didn't like holds true for most children. If your cousin's girl didn't like the Barbie, she probably wouldn't play with it.
The whole point of childhood is having an idea of "I" without having a strong idea of identity so that we have a solid reference point for learning about the world without experiencing the blind spots that ego identities often impose. Consequently, we demand what we want and refuse what we don't want without thought. We rarely make compromises and our needs and wants cannot really be reasoned away by others (because our needs and wants in childhood are more instinctual/subconscious/impulsive than they are logical and experience-based).
I think we are most free to choose in childhood. By adulthood we have developed functioning identities and feel obligated to stay consistent with our idea of self. Many people take portions of their identification from a society's expectation of them, which is when freedom of choice becomes heavily blurred.
I think what is important here is to give children different options. I personally got a lot of toys from my family that I didn't use that much as a kid, most notably sports equipment. Meccano and (later, as it was expensive) Lego on the other hand.. Not to mention computers, that was love on the first sight.
Of course radical feminists reject all this evidence en masse and pretend its all a social construct, which is absolute nonsense.
To add to your rhetorical question of the inability to trace the source of gender preferences, some research on non-human primates was done.
In 2002, a study of vervet monkeys in the UK found male/female preferences for boy/girl toys.
That behavior was independently replicated in 2008 with rhesus monkeys in the USA displaying the same gender preferences.
There were also other studies of human infants displaying gender preferences, and studies showing similar preferences across different cultures and countries.
Those results will probably not change minds on either side because -- the belief that it's mostly nature or belief that it's mostly nurture -- is too ingrained for any evidence to modify.
If one leans towards innate differences, the studies confirm the beliefs.
If one leans toward cultural influences, then the studies are rejected with "monkeys are not humans!" or "who's to say the lab monkeys didn't learn boy/girl toy preferences from the researchers?"
The meta question then becomes: is it even possible to construct a nature/nurture science experiment that can actually convince one side or the other?
(The answer seems to be "no" based on several hundred years of debate on "nature vs nurture" with no final resolution.)
 2002 paper: https://www.ehbonline.org/article/S1090-5138(02)00107-1/abst...
 2008 paper: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2583786/