I think there are some weird notions going around in this thread.
1) People overestimate how much agency children have over their opinions by a lot. As a parent, you can tell your child pretty much anything you want and your child will copy that belief. A child's life depends on their parents. Even children who are being sexually abused will side with the parent who abuses them, which is the number one reason why child abuse is so horrible and has so many terrible effects even decades later. To use another example: A child knowing what a boy is and what a girl is (and what typical gender stereotypes are) doesn't mean the child has any idea what it means to be the other gender.
2) This can be abused both ways. Parents pushing children towards seeing themselves as transgender abuse their child. And parents who refuse to allow their child to express themselves also abuse their child. Of course, since this is a political discussion, the usual blindness on the left / right eye pops up again for a lot of people and the problem is seen in a way too one-sided manner.
Personally, I think children should express themselves in which ever way they want but we should keep hard labels out of this as much as possible.The human body only fully diverges into the male / female dichotomy once puberty hits. Men and women are physically way further apart than girls and boys are, so children cannot fully grasp what this means. So in this regard I very firmly agree with the political right. I agree with the political left, though, when it comes to fully expressing yourself. If my child insisted on wearing clothes made for the other gender and this wasn't caused by me or my wife's political opinions, why would I refuse my son / daughter that wish? My main point is that things should stay playful and we shouldn't put hard labels on children if it can be avoided. Children are children, let them not worry too much about all the crap in life as long as they are young. I neither agree with conservatives who tell their child "girls / boys don't do that" nor progressives who prematurely want to label them as transgender.
I've seen a few stories of transgender couples having "transgender" children - what's the mathematical chance of both parents and their child to all be transgender? This is probably just the case of a child emulating their parents. That being said, as long as we are not talking about hormone therapy for children or manipulating your child into seeing things this way (which, unfortunately, some of these pattens probably do to get media attention), the child can probably grow out of this if his/her parents aren't pushing for anything and the child happens to be among the 65% of children who don't turn out to be transgender in the end.
So in the end, the discussion is blown out of proportions a bit from both sides I think.
