From the tribes of the Southwest (see Williams' 2005 work The Berdache Tradition) to Europe's Chevalier d'Eon, transgender people are a cultural universal. The science is not well-understood. I suspect that there is a neurologically-hard-coded disconnect, perhaps due to uterine hormone exposure during fetal development, between what the brain expects in terms of bodies/hormones and what it got. Either way, our brains are the root of who we are, not our bodies. It's just common sense to help people resolve this disconnect, instead of demonizing them for it. 

But, bodies do matter for sports. I would have no problem with sports leagues that are formed not based on gender, but on some combo of weight class and skeletal structure? Honestly, it seems like it would produce fair competition and more interesting games. At a practical level, however, I get that lockerooms are currently categorized by gender. Just build in more privacy (and some security options). Think about how much school-based lockeroom bullying could have been eliminated with basic ways to inform adults about what was happening?  

Now, the elephant in the room is whether some kids might pursue biological transition based on a delusion of being transgender. Certainly, it's possible. Kids have limited experience of the world. Some are foolish enough to think that the gender roles they experience in their particular culture in their particular decade are somehow universal. A half decent psychologist can sort out true transgender kids from those who are merely confused. It takes time, sure. If we really care about kids, we'll make sure they have access to at least half-decent psychologists. 

It's all just common sense.