No one wants children to be sold into servitude. However, the state impinging into family affairs to the point where it can be 100% prevented is essentially making the state the full de-facto guardians of the children. This is the state of affairs where children are informing on their parents to the state.

It's in part up to the parents in a case like this, to determine if what they're doing is selling their children out. That should be a matter of relationships within the family. (Government should be supervising how such business treat and house children. It shouldn't be impinging itself on the relationship between parent and child for any but the most extreme circumstances.)

If so, there's no point debating with you - I find the practice morally reprehensible

Using such debate tactics is dishonest on a few levels. 1) The likelihood that you are talking to someone that reprehensible is rather small 2) The likelihood that you are not exercising the principle of charity and imposing unnecessary emotional toxicity in a nuanced discussion (to your own short-term rhetorical benefit) is rather much larger.

and yes, the systems that make such an arrangement in any way palatable are also reprehensible

There are and always will be a few horrible people who will do that to their own children. Giving over tremendous power to the state over the intimate affairs of absolutely everyone for the sake of preventing rare instances strikes me as hugely unwise. Imagine what the plight of gay teenagers in the past would have been like in an unfriendly regime of such power, where state power reached into family affairs. (Or the plight of such teenagers in totalitarian theocracies today.)

As someone who was a child who was sent away to school by his parents, let me tell you that they matter a great deal. (Particularly from the POV of an adult looking back vs. the experiences of a child.) They are precisely the questions that should be asked to distinguish manipulations of "Think of the children!" hysteria and genuine issues of the human rights of minors.

Is there any answer to your questions that makes indentured servitude of minors acceptable?

Of course not. But I can imagine the level of state power and impingement on private family life needed to prevent that 100%, and that would be quite horrible as well. Government should regulate the conditions under which children are kept and raised by schools. However, the degree to which someone else should decide for parents what to do also needs to take into account the rights of parents and families.

I understand what it's like to be sent away to school by one's parents to experience significant hardship. It's not a matter where one should cry out for more state intervention, lightly.