Seems to me the issue is less about public schooling than it is about the government *requiring* attendance until age 16 or whatever. Without mandatory attendance, school is an opportunity. With it, it's an obligation.

But it's not even close to slavery, in part because the student is not providing labor that produces labor or a service. Perhaps he or she does so indirectly, through the compulsory consumption of a service (education) that is supplied by an industry (the education establishment) that has no other purpose or constituency. But that's not slavery.

Now, you could argue that elementary education rightfully falls within the public sphere of compulsion, like traffic and littering laws, or public health policy. After all, those who do not get at least a minimal education are more likely to be a burden on society, or even outright criminals, than those who get one. And minors are generally recognizes as having few if any rights, and to be less capable of making informed decisions such as whether to attend school.