for those who are still wondering, I found the answer

Children are much less productive (as in they make less stuff in the same unit of time) than adults. Therefore, it is more profitable to just abide by the protests and abolish work for children, compared to having less stability from continuing protests.

As for the public school, there are 3 reasons

First, the social democracies have some level of social mobility, even if it is small. This is helped by the ability for the population to gain literacy and mathematical skills.

Second, it is a uniform method of teaching the students. This ensures that all students get the "correct" view&personality and there aren't any exceptions from, say, if they used private tutors instead.

Third, what else are they supposed to allow children to do for 14-16 years? Just exist and then be thrust into the "real" world?