Once upon a time many years ago, bullies used to get punched in the nose. This taught the bully not to be a bully, and the bullied to stand up for themselves. The puncher nor the bully told a grown up, because you don't snitch. The situation was handled. End of problem.
Now thanks to Oprah, Dr. Phil, and so called modern parenting we make entire mountain ranges out of the tiniest ant hill problems in our kid's lives.
Now kids won't punch a bully because they are taught to talk about every problem they encounter. This gives the bully even more power. Then if a kid does knock a bully on their butt, the kid is in trouble because of some asinine zero tolerance policy. Then the parents of the bully and the bullied show up at school demanding that their respective kids needs and uniqueness be catered to. Then the adults get into a huge self centered convoluted argument, and end up more concerned about being right, instead of helping the kids.
The kids in the meantime, are getting a ton of mixed signals.
The bullied are told not to defend themselves, but try and figure out why the other person wants to bully them, and see if they can change their own behavior or, talk the bully into not being a bully. If that doesn't work, run and tell the nearest adult to save you. So what's the kid supposed to do when he gets punched again in mid-sentence of trying to have a meaningful dialogue with his bully? What if there is no adult to run to? After being rescued from the bully the bullied are told that they now have post traumatic stress disorder as a result of their ordeal, but some therapy and this prescription will help with that.
The bully is not told he's a jerk and he needs to quit acting like that before someone kicks his butt for him. Instead he is put into therapy where he is told there's nothing wrong with him, he's just expressing himself in ways that others find difficult to understand. Along with this positive reinforcement, the bully is given a prescription to help him better cope with the stress and anxiety of being a bully, er...I mean kid.
Now we've got medicated, issue laden kids. They've learned no real independent coping skills, or anything about natural consequences. Since the bullied are made to feel even more powerless by these circumstances, they become an even more attractive target for bullies. This cycle continues until one day they reach a point where they can't take it any more. Then we have suicides, gun violence, or whatever extreme measure the kid decides to go to in order to feel like he's handled the situation on his own.
My kids are taught to ignore verbal bully BS entirely. My ten year old had a blast doing this because the ignoring annoyed the heck out of the bully. Eventually the kid got frustrated and left my son alone. If someone takes their personal property, my children have my permission to recover their property by any means necessary, within reason. We have discussed what within reason means. They face serious consequences for going overboard or not getting an adult if the situation escalates. If someone physically assaults them, my children have my permission and encouragement to respond in kind. Again, we have had many discussions on what it means to go too far even if you are in the right. My children face very serious consequences if they are ever the bully or the starter of a fight. However, regardless of the consequences they receive at school, my children will never be punished in their home for refusing to be a victim, and handling their problems independently.
I'm not encouraging a free for all on the playground. I just don't feel that we give our kids enough chances to grow and mature by handling conflicts on their own. Obviously if a situation escalates, we as parents need to become involved. That involvement should be minimal and help, not hinder the growth of the kids.
How to handle a bully is only one of many common childhood experiences that so called modern parenting has gotten wrong. In my opinion the adults who grew from generations where there wasn't a diagnosis for every problem and a pill to fix it, are stronger and better equipped to handle struggle and adversity than our current generations are going to be. We have gotten so focused on teaching children how unique and special they are, that we've forgotten to teach them how to be a part of a society. All prosperous societies realize that "the needs of the many outweigh the needs of the few, or the one." Society today is telling kids that their uniqueness and their personal needs and wants are what is most important.
I am proud to have been raised by an old school parent. I call her all the time and thank her for being the meanest Mom in the world. I am proud to be an old school Dad. I spanked, I yell, I curse, I ground, I ask questions and demand more than one word answers. My kids are happy, strong, self sufficient, relatively issue free people that will not be helpless victims in life.
This turned into a bit of a rant, but I feel strongly about how poorly kids are being taught to handle things. I guess my answer to the OP is, no I would not have done the same. My daughter would have hit the girl and told her that's what she gets for biting. If the problem escalated after that, then my daughter would go get a teacher. The fact that my daughter tried to handle the situation herself would have kept the other kids from calling her a snitch, since she only got the teacher because the bully escalated the issue. Either way, the bully knows that biting my daughter is not a good idea. However, by trying to handle things herself my daughter gains the respect of her classmates and possibly the bully, instead of being ridiculed as a snitch. I'd rather have my daughter in trouble with the principal but feared by a bully and respected by her peers, than for her to be a powerless victim.
I don't expect many people to agree with my stance on this issue. To the OP, please do not take this post as a personal affront to your parenting choices. That is not how I meant it.
Would I do the same? No.