The problem isn't trans kids transitioning, getting puberty blockers, etc. That's not actually what the republicans are upset about. They don't care about queer kids or what happens to their bodies.
The problem is that THEIR kids will go to school with them, and might learn to love and respect them. And they can't have that. They have to make sure their kids don't see trans kids, don't see drag queens. Their kids can't get the wild idea that queer people might be people, and even worse, their kids can't get the idea that they might be loved and supported if they themselves are queer.
It's not about the trans kids, it's about their own kids being disgusted at their parents' bigotry and accepting people for who they are. The only way to fight the fact that the youth don't support them is to keep them as ignorant as possible.