On my motivations, society does allow bullying to happen, and the discrepancy between what it allows to happen to children in the context of bullying, and what it allows in most other cases, requires an explanation.

This requires putting oneself in the shoes of the people responsible for this situation, including people who minimize the importance or harm of bullying, people who justify it outright, and people who silently ignore the issue.

In doing so I am going to argue that given certain assumptions (both positive and normative), bullying should be permitted or justified. That is the inevitable outcome of being intellectually honest.

You claim there is a difference between upholding social value through reprimand, and bullying. I couldn't tell if being "taught a lesson" referred to violence so I won't assume either way. If it didn't refer to violence, then in an individualistic society like ours, people could just ignore it. Maybe in Scandinavia a verbal reprimand is enough, but not in the rest of Europe or the English speaking world.

And yet a free for all of violence would not work either (which seems to be the straw man of bullying you are using in your argument). Bullying is not only violence and the establishment of a pecking order, and also not only punishing violators of the social code. Rather it is doing these things together in a complementary way. The social code is respected because it comes from the most dominant people, and the violence is justified because it also helps cement the social code.

This explains why schools and prisons punish people who defend themselves against bullies more than the bullies themselves. To defend yourself is anti-social violence. Furthermore, as students get older, this social order morphs into the actual social order of the adult world. It is not only a pecking order that teachers tolerate because it makes the school easier to manage. It also gains the gloss of society's actual moral code.

For example, students might be bullied because they exhibit middle class speech and behavior, and are unwilling or unable to adopt "cool" working class behavior. A teacher might be sympathetic, but also consider that without this kind of bullying, middle class student could be as snooty as they liked, and that forcing middle class students to respect working class students (even out of fear) would help societal cohesion in the bigger picture.

The values that bullies enforce are arbitrary, but not entirely arbitrary. If they only punished the truly snooty, or the truly degenerate, would progressives or conservatives respectively complain? I see many people encouraging bullying tech workers in SF because they claim those workers really don't respect the values of the communities they live in. And (this was a while ago) I read a Time magazine article about the bullying of the Columbine shooters, which claimed precisely that their bullying was justified because they violated social norms (as the article said quoting a footballer "they were a bunch of fxxxxts", and this was in the 1990's so the implication wasn't that the footballer was in the wrong!).

I am not saying that society will fall apart without bullying, but that people who want to maintain social norms that aren't enforced by law, logically have little choice but to accept that experiencing an environment with bullying is an imperfect but necessary precursor to knowing your place in society.