As others have said in various ways, violence in some capacity is necessary to deal with bullies and abusive people in general. Non-violent methods simply enable or encourage the predatory habits of those kinds of people, so they're not a viable solution. As barbaric as it may sound, the question should not be "is violence an acceptable answer to violence?" it should be "how much violence is an acceptable answer to this instance of violence?" It all comes down to a matter of proportionality and severity, and it varies a lot from case to case.

A kid roughly shoved another kid? A firm swat to the head will probably suffice.

A kid beat the crap out of another kid? I don't care what modern coddling parenting nonsense says, that deserves some ass whooping with a belt.

An adult hit another adult? Return it to the attacker with interest and call it good, they're old enough to know better. Exceptions to this include if they injured the person (ie more than just pain and bruises), or they're attacking a defenseless person, or it's a repeated thing where they didn't learn from past times getting punched for hitting someone else; in any of those cases you can slap their shit and also call the cops because they deserve to get their shit slapped legally as well as physically at that point.

An adult physically abused a child? This is one of those situations where if you just trade violence for violence and leave them be afterward, odds are very good they'll later go and vent their ire on the same target and that's no good. Luckily it turns out that whooping ass isn't the only kind of violence, and that's where jail comes in. Throw them into jail for a few years to cool down, slap a felony child abuse charge on their record, and bam, you've just beat the shit out of their entire life without needing to resort to traditional physical violence (though there's the fun bonus that child abusers and molesters tend to get their asses kicked regularly in jail).

All of those should be coupled with some of those non-violent methods, by the way. All that counseling and explaining why it was wrong and how they should feel bad and so forth should come during and/or after the administration of violence. Alone that stuff does little to nothing, but coupled with a fresh reminder of the direct penalties of further transgressions it might actually work. Violence to answer violence is not pretty and it's not a perfect system, but it's a hell of a lot better than the non-violent routes.