The current WPATH standards of care basically cite studies which said that of gender-nonconforming children admitted to therapy, only about 10%-25% had their transgender identity persist into adulthood. A higher percentage were likely to eventually identify as gay. Once they reached early puberty, though, 70 out of 70 who took puberty-blocking hormones had their transgender identity persist into adulthood. So once early adolescence is reached, and body aversion is there, it's not going away, you're trans, and any delay in therapy will just make it worse.
This same section does say, however, that those with the strongest gender-nonconforming feelings as children, those who play completely with the toys of the opposite sex, demand that they are a member of the opposite sex, and show early signs of body aversion, are more likely to have their trans identity persist.
So I can definitely approve of childhood gender transition if they do indeed show very strong signs that they really do mentally have a cross-sex gender identity, to the point that it causes them emotional distress. But until new scientific evidence proves otherwise, I don't think we can necessarily claim that all kids who think they might be the opposite sex are trans. Kids don't really understand gender. I know I didn't as a kid. I still thought that anyone who had long hair was a girl, and anyone who played with girls' toys was a girl and anyone who wore girls' clothes was a girl. I used to think that I could change into one if I did these things. So I don't think that a lot of kids can tell the difference. They can't quite mentally separate physical sex from gender expression yet.
Once these kids reach about 11-12 years old, though, and show clear signs of body aversion, which is a pretty definitive sign that their brains are hardwired to be the opposite sex, it needs to be spotted and treated as soon as humanely possible.
I had no idea I was trans as a kid. I didn't have that much gender-nonconforming behavior, at least nothing that would have gotten me noticed. But once my trans identity and body aversion started developing around age 12-13, it hasn't changed since. And it only got worse and worse as my body masculinized more and more.
(And yes, that video that someone posted about trans youth does make me VERY dysphoric. Especially the part with the twins, because you actually get to see how much of a difference blocking the male puberty makes. It's like "here's what you actually did go through, which now you can never undo, you hideous unlucky unfixably-masculine person, and here's the beautiful completely-female person that you could have been if you'd paid more attention to your jealousy of girls and hatred of your masculinizing body at that age." Urgh...

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