Sometimes this is probably legit, e.g. the state will take children who have cigarettes put out on their foreheads, but sometimes it's not legit, e.g. to make the children of poor people go to public school because children are not allowed to stay at home if mom and dad have to work, so it is de facto custody by the state for poors.
In the UK, of course, poor children only go to public school if they get a scholarship
That aside, in the UK, the state does not force you to send your child to school, though it does insist that she is educated. There are lots of home schooled kids out there. It doesn't
have to be your home, or you that does the educating.
Furthermore, children below a certain age are not allowed to be left unattended
for the same reason as you can't put fags out on them, both being abuse. If child protection laws amount to state custody of children, are you also arguing that the state has custody of animals because there are laws preventing cruelty to them?
And finally, the fact that when the state does take custody of children it goes through a lengthy legal process which the parents have rights in shows that it does not see itself as having custody as of right.