For each sub-adult child, somebody or some body will determine how that child spends the time between age zero and age 18.
It's not that simple. A child's activities between the ages of 0 and 18
1 are chiefly directed by their parents or guardians (within the law that restricts the amount of time they can engage in paid work, and within the law that specifies that they must receive an education). The most the 'bureaucrats' can do is offer education options in the form of publicly funded schools between the ages of the school starting
2 and leaving age
3 for something like 6.5 hours a day during the school terms, and within that offer of public schooling they have to take into account the wishes of the communities as to how they want public education to be funded and directed. That's democracy.
The public policy alternatives are not between flawed parents, on the one hand, and intelligent, informed, altruistic bureaucrats, on the other, but between flawed humans on both hands. Some parents will waste the human potential of their children. Some bureaucrats will waste the human potential of thousands of children and millions in taxpayers' money. Abundant evidence supports policies that shift power from State bureaucracies to parents.
Again, it's not that simple. Just as you accept that some parents and some bureaucrats are 'flawed' by which I guess that you mean they do not put the child's interests first, so the majority of parents and educators (teachers and policymakers) are 'intelligent, informed and altruistic'. They do put the child's interest first - their interest to receive a rounded education tailored to their needs and ability, and giving the children in their charge a safe environment in which to develop their full potential.
If the majority of teachers and educators are not intelligent, informed and altruistic in the USA school system, the answer is to hire people who are all of those things. If the public school system, which will always be educating the majority of children in every modern democracy, cannot offer that rounded education, tailored to the range of needs and abilities of children and providing them with that safe environment, the answer which makes economic sense both for the present and the future is to improve that public school system.
The choice to opt out of the public education system is still there for parents who can afford private schooling (here in the UK, around half of children in independent schools receive some sort of grant or bursary from the schools to help with fees and equipment which can cover just a portion or the entire cost), and the choice to homeschool should still be there for those parents who wish to do that.
Even in publicly funded boarding schools (there are some in the UK), the parents or guardians still have much more influence over the child and how they want that child to be educated, simply by virtue of the fact that for the years prior to them starting school and during the holidays the parents are in charge, and they choose the school they want their children to attend.
Investigoogling has not helped me understand whether parents have a choice about which publicly funded school they can send their child to in the USA. If they do not, then that is another area which could be considered in order to improve the public school system.
Over here in the UK, parents can apply to any school they wish, although they may not get a place depending on their priority.
4 State schools are in competition with each other for pupils and the funding (which follows the child), and different schools will have different ways of teaching within the National Curriculum, different policies on teaching children up or down out of their age group, streaming by ability, the way they help children with special needs, and whether they emphasise science or the arts just to name a few.
1. Age of majority may vary by country
2. School starting age may vary by country
3. School leaving age may vary by country
4. Priorities for places depend on many factors, whether the child is in the care system, how far away they live from the school, whether they have a sibling already at the school, just to name a few. If a child does not get a place at their preferred school, there is an appeals system.