> You mean they enroll back at a private school to get their education?

I mean the money that government wastes keeping them in school while they are 15 and don't want to learn, can be given to them later when/if they decide to learn.

> most likely the path is that they just stay at home doing nothing all day if their parents let them.

That's up to the parent to decide: leave them at home, convince them to find a job, go to special school or a class for misbehaving children, go to trade school etc.

Those who turn to vagrancy/crime do it anyway, as they have enough time outside of school too.

> child has to be in education until they turn 18.

> employing 15 year olds is [not] legal

These are not physical laws given to us from above, these are rather misguided attempts by politicians to look good, and are harmful to the society.

Imagine that instead of prisons we were forcing criminals to go spend time sitting in offices and disrupting normal work. What we do with children now is equally effective.