The real question is why does the state have any say? There's no fundamental reason to trust the state over a child's parents; the reason the state has any say has more to do with might rather than right.
The reason is because the state has an obligation to take reasonable efforts to ensure the basic human rights of all people within its jurisdiction are protected. That is, the state has a duty towards the children that is independent of the state's relationship with the parents.
There also is a fundamental reason to trust the state over a child's parents: the state has considerably more transparency and oversight over its actions than the parents do.
There's no fundamental reason to trust the state over a child's parents.
Most people don't want to harm children and won't sit by when they see it happening. They want to help. That means it's better to get more people involved in the lives of children. "The state" is one way we codify that.
I think there's a good argument for enabling communities to assume that role rather than central government, but some group should be there to protect children from neglect.
Some parents clearly aren't fit to be parents at all, I trust our collective wisdom as embodied by our government a lot more. Even though child protection services are horribly underfunded and have also become quite bad.
Because Europeans believe more in collectivism, pushing for greater good together for the society. Also compared to Americans who tend you like individualism we are not paranoid of our government because they are part of us, even if we don't like the politicians. Two different mindsets, it's hard to argue with that libertarian lens with Europeans.
In Germany, I have to tell the state where I live and every time I move, I have to tell them again. I can't open a bank account, get health insurance, rent a proper place, I can't do anything, till the local council can be bothered staffing the registration office enough to see me. I certainly live at a certain place, but it's as if I'm staying at that address on holiday.
In Australia, I don't have to register with the state. If I move house, I have enough things on my mind. If a person is born, died, gets married or (sometimes) changes their name in Australia, someone has to report that fact to the government. But a person is a person, or dead, or married, or has a different name while they're waiting for the bureaucracy to do its thing.
In both cases, children have to go to school. Even in the absence of any registration or reporting, there's still an obligation. So what does the registration actually achieve?
The government has the power and ability to find out where a person lives. Why compel me to do something you can make obnoxiously difficult? - and which serves no apparent purpose?
Or are you suggesting it is okay to have children and never educate at all?