Children should be educated and protected by their parents and family, not the State. First, because very kid is different and it's unthinkable to create a catch-all solution that is good for everybody. Second, what is the State good at? Anything that is done at the government level is about 100 times worse than private businesses, and 1000 times worse than what the vast majority of parents can do. You don't want the State deciding what you are allowed to look at on the internet, if anything they would do more damage than anything else.
I also don't get why people have to freak out about everything nowadays, as if kids are completely helpless against the outside world. When I was a kid there was all kind of extremely dangerous stuff going on, and as kids we got warned by our parents and everyone is still alive.
Not every child has this though. Parents vary greatly in competence and ethics, and virtually every society and government recognises children as having rights separable from their parents' and draws a line somewhere as to what parental conduct is acceptable.
In addition to not being granted every responsibility, parents often don't want every responsibility. Most people in western societies expect the state to provide education, and some level of public security.
That said, I agree that the UK takes this to an authoritarian extreme.
In the extremely rare cases where parents aren't able to take care of children, new guardians might be assigned by the government, but then they would take on the same responsibilities as the original parents, including not letting a kid starve himself to death because of an internet video.
If you mean that the State is justified to intervene in some cases sure--but it depends how extreme these cases are. Certainly not what people post on the internet (on top of already existing laws that regulate dangerous behavior).
The idea that parents should have such sole responsibility over their children is one extreme of a spectrum of opinion. UK government policy tends to be on the other end of that spectrum, with the opinion that children are the state's responsibility, with forced adoption being used much more readily than the rest of Europe, for example.
It's also hard to reconcile the idea of childrens' mental health being the sole responsibility of parents with the causes of childhood mental health problems very commonly being abusive and neglectful parenting.