Because it's very easy for the creeps already thinking of your children to paint these rejecting this type of the laws as those who want to see children hurt.

Regardless how stupid this argument is, rags will always pounce on it.

This is just a dirty trick of the creeps to make the resistance harder.

I think it's because, without further context, it's so hard to argue against. Pretty much every person in every culture cares deeply about their children. So if you can successfully hitch your position to that idea, it too becomes hard to argue against.

It's the same with tough on crime. "What, you want criminals to keep getting away with it?!"

Because adults remain children. As in, their parent’s kids and therefore property. [edit: I should mention also property of the state beyond that] It’s less explicit in US I guess but in some places that’s very blunt - if you don’t support your parents enough you can be sued for abuse. And there are situations where an adult in us has been declared too irresponsible and forced into conversion camps by parents in the US. It’s insane, yes, and if you’re lucky enough this might be entirely invisible to you. But if you’re gray or trans or autistic and get a but unlucky this can become a very harsh reality.

Protect the children refers to a type of property, not a type of human.

Because it is the moral responsibility of adults to care for not just their children but all our children. Occasional surrendering of rights is appropriate in that endeavor.