Hell no. It should not be mandatory and state run.
It should not be mandatory at all. If they take up working on cars and do not want to get an education, this happened with a kid I once knew and he was successful with it, then good for them. If they fail and have to work at burger king the rest of their lives I am fine with that also. I am fine with state run, but we should switch to a voucher system and make public schools compete.