I think it's extremely important for trans* children to be able to live how they know is right. And believe me. Kids know who they are by the age of ten. Doctors and psychiatrists have evidenced that a child's inner sense of gender often begins to emerge by the age of three.
I think the problem is some people freak when they hear 'hormone therapy' and 'children' in the same sentence. But that doesn't JUST mean estrogen or testosterone. Gender-variant kids who haven't hit puberty yet and aren't yet at the age of their post-transition gender's pubertal group often go on hormone blockers until the doctors can confirm that the kid really knows what zie wants. It holds puberty off for as long as they need.
But the people who say that the child should be forced to endure the absolute horror of going through puberty in the wrong body? Why? Why would you put a child through that? Would you want to go through the puberty of the opposite gender to what you know you are? I don't think so. Would you want to feel your body rebelling against you at every step? I doubt it. So why would you force that on another person?
Especially because some of the secondary sex characteristics (facial hair, breasts, etc) are very hard to remove and set right! Not to mention traumatizing, expensive, and grueling!
If an FtM child goes on blockers and then starts hormones for the gender they will be transitioning to BEFORE puberty, they avoid having to have top surgery to have their chest tissue removed. They avoid having female-width hips for the rest of their lives. They avoid having to deal with the trauma of bleeding from anatomy they know they shouldn't even have!
If an MtF child goes on blockers and then starts hormones for the gender they will be transitioning to BEFORE puberty, they avoid having to have facial hair removed via laser. They avoid having to develop a height above that of the standard female range. They avoid having to have their genitalia rebelling against them in terms of development!
And most importantly, this: If a trans* child transitions at a young age, they get to have the childhood they deserve. As their correct gender. Personally? I'd kill to have had a boy's childhood. But I'm past that phase and I will never get that back.
Would you really force a child to suffer through all of that rather than being happy as their right gender?