Bullies need to be identified as simply immature, treated as children that have not graduated to their age. That really impacts the individual. Make them wear identifying clothing as a "special case" and they will mature very fast.
There are lots of reasons this stuff happens, but one of them is definitely that some kids aren't acting out for school reasons but for attention from their parents.
Classical stereotypical case is that the bully himself has an abusive/alcoholic father. There's a lot of complexity in what's involved here, but society is only equipped to deal with the "immaturity" case which is real but not the only story.
While it is possible to stamp out physical bullying, psychological and verbal bullying are near impossible to eliminate, and so any sizable school which denies having it is lying. It is a matter of degree, and how they handle it.