"The only thing which might be said is unproductive with kids' childhood is the long summer breaks"
That assumes that schooling is relevant to a kids future. My 4th grader already operates at a numeracy and literacy level higher than adults "are supposed to". You're not supposed to be able to calculate percentages and figure out word problems, that will ruin the employment of tax preparers. You're not supposed to be able to read Harry Potter, you're supposed to watch TV. Achievement at or above a 4th grade level is strongly discouraged on a mass cultural level, although obviously not here on HN. Vocational training is useless because he won't be in the workforce for a decade or more making "training" useless, and according to labor force participation rates the odds are reasonable he will not be a participant in the labor force anyway, especially as trends continue, making vocational "education" useless. Depressingly, there are a lot of white collar jobs that boil down to adult day care, fundamentally, where lack of productivity is probably a net societal gain!
So... the true cost is free day care (aka school) is not available in the summer. On the other hand educational costs aka property taxes are microscopically lower. Obviously we are not going to permit teachers to experience a 30% boost in standard of living were we to expand their workload into the summer, so labor costs will not change but obviously energy costs to keep the lights on, and probably air conditioning, would increase, along with internet bandwidth use at the school, cleaning chemicals used, trash bags filled, etc etc.
The reason why kids "need" ever more and more years of education for ever simpler jobs (on a societal average...), is to keep them out of the labor force, to reduce competition. Otherwise, if bright teens put all the adults out of work, the adults wouldn't be able to afford children, which breaks the cycle in about a generation. You need to keep the adults employed in their breeding years long enough to squirt out the next generations kids, no need for much more, and keeping younger adults / teens out of the workforce is a social engineering technology to accomplish that. Its quite possible as existing trends continue that we'll naturally assume no one would ever hire a supermarket cashier or waitress without a PHD in something, perhaps philosophy. That would serve a useful purpose of keeping as many people in (profitable) schools as possible, maximization of student loan income, etc.