There is a massive leap between "let them bully other kids" and "we have to cane them" and pretending like only pain is the solution, especially in case of children where bullying is often a second order effect, is sick.
These rules should be implemented locally at a town or city level. No need to enforce the same set of rules across all society.
And it's interesting you bring up that bullying is a second order affect. If one of the parents is abusive, that should be something that has physical consequences. Solve the problem at the source, stop wringing our hands and getting lawyers / police involved for everything. That's not scalable and as a result there are a bunch of unsolvable problems in our society today.
Like I dont understand what you're saying at all because it seems like you want the social contract but also give anyone the agency to conduct violence and both cannot exists at the same time. We live in communities and created the police and law precisely because personal grudges and fights cannot scale and work to be a functional society. God i hope you are trolling
An example that requires police to be involved: Small Town A has a law stating that anyone dealing drugs must be caned for the first offense. Someone deals drugs in Small Town, so police catch them and cane them.
I swear, some of y'all just dream of being able to cane people or something.