Actually the adult criminal justice system does offer extensive opportunities for education.
Most child offenders come from broken homes and are not lost causes. The problem is, they do not have stable environments to return to once released. They are often sent back into the situation from which they came, which is usually abusive or neglectful, and succumb (or forced) to the pressures of very powerful gangs, or become addicted to drugs at young ages. Some of these children have never had a stable home in their life, being moved from foster home to foster home, and many of them have been abused physically and/or sexually by multiple people who were supposed to protect them.
If these children were instead placed in a stable, safe environment with good parental influences, chances are they would grow up to become productive members of society.
The education these kids received while in the system does make an impact. At the very least it presents these kids with other options in life, and occasionally, the very strong ones or the very lucky ones are able to turn their life around because of it. I had a family member who successfully beat the statistics due to one such educational program.
Believe or not, the country with the most succesful juvenile justice system, which has the lowest rate of repeat offenders, is Turkey.
In Turkey the children are not locked in all day. They leave the facility during the day to attend school or work, and return at night. While crime among juveniles in Turkey is more of an economic issue than a social issue, I believe such a system would have great success in the U.S. as it literally pulls the children from the negative environmental influences without isolating them from the real world, all the while showing them what they can do to better their lives.
Your idea of just giving up on them will do nothing positive.
I've known many of these kids.