The gender binary system we have in place at the moment - that men are men, women are women, everything else doesn't exist etc. - is an ancient social construct that's as deeply rooted in labels and pigeonholes as the more more fluid system that's being proposed by transgender activists today. The old binary system might have been around for centuries but it doesn't mean it wasn't invented and enforced. For example, Indigenous and Native North Americans were perfectly happy with people in their society being "two-spirited" (the spirit of a man and a woman inside the same body) and often practised gender swapping in many different forms - at least until Christopher Columbus rocked up on their land and forced two-spirit people to conform to more conventional European Christian gender roles or face death. The rest is the gruesome history of the United States of America.
Gender, biology and sex not necessarily being fixed to each other, and gender being a spectrum as opposed to one thing or the other, isn't a perfect system, but the world doesn't have to live by it every day. The binary gender recognition system we have in place now is adequate if problematic and works best for the majority - as much as I'm an advocate for transgender rights and want to see a society where gender fluidity is accepted and encouraged, it's still perfectly acceptable and practical to assign a baby's gender at birth based on their genitals. What's not acceptable, however, is what happens afterwards, where primary socialisation is impacted so heavily by rigid gender roles that any child who grows up with gender dysphoria feels trapped, isolated and ignored, and then ends up as a social outcast (or the subject of ridicule from angry men on football forums) when they try to do something about it, like beginning hormone therapy before biology takes a hold of them.
Transgender people of all shapes and sizes have existed in thousands of cultures and societies for centuries and it's high time we recognised that and started treating them like human beings again. And the best first step for some people would be for them to see stories like this one and simply live and let live. Boys want to dress up as Disney princesses and wear pink, girls want to join football teams and play with cowboys and dinosaurs, and when they get older they might want to begin hormone therapy so that their physical appearance better resembles the person they believe themselves to be on the inside - so what?