The problem with your typical stand up martial arts to deal with bullies is twofold. One is the zero tolerance policies in schools and two, what I call the "aliveness" problem. I'll deal with one problem at a time here.
The first problem arises when you think you know how to fight, but can't because of school policy. There are consequences to defending yourself with a zero tolerance policy, so the kids who typically get bullied continue to just put up with it to avoid getting into even more trouble.
With traditional martial arts, defensive techniques use striking a lot of the time (not always). When you have to use strikes to defend yourself, there is no way to ramp up the escalation, you either punch somebody or you don't. The possibility of really hurting someone using these techniques is extremely high. I don't think any kid, even when defending themselves, wants to hurt anyone in a bad way.
Not only are you putting someone else in danger with striking, you are at risk of hurting yourself as well. Even if everything goes your way so to speak, the possibility of breaking your own hands are pretty high. If you are in a fight where your only weapons are your fists and feet, both parties are going to take significant damage. That's no fun even if you win.
So, the moral of the story is, if you use fisticuffs to stop bullying you will both probably get hurt and in trouble to one degree or another. Don't forget about getting sued for any permanent damage you might do as well.
The second problem with striking for self defense from bullying is the aliveness problem of training. You can't really spar (play fight) with full contact, so kids are never really sure what will happen if they ever really hit someone as hard as they can.
The self defense learned in a typical martial art is taught with a willing partner, so you really don't know what will happen in a real fight. Trust me, this is not lost even on young practitioners. It's hard to have any faith in your abilities if you've never really tested them yet.
I know all this personally because I was always bullied in school even though I had a black belt in Tae Kwon Do by the time I was 11 years old. My parents put me in martial arts because I kept getting my butt kicked by one of my neighbors. I actually feel I got pretty good at all the moves and the katats and what not, but I was still always too scared to defend myself for fear of it not working in real life or the greater possibility of getting in even more trouble.
Many kids are told that they can defend themselves if they need to but still feel powerless to do so. That's because they don't have a self defense toolbox they feel comfortable with. It's like telling a kid to build a treehouse with no tools. If I'm not comfortable with my ability to do something, especially something with physical consequences, I'm just not going to do it.
That's why I've become a huge fan of Brazilian Jiu Jitsu. This martial art teaches self defense in a way where you can control a fight without hurting someone and where you can train with a partner with 100% effort without fear of hurting each other.
If a kid who gets bullied simply controls someone else while they negotiate with them, they are much less likely to get in any kind of trouble. In this manner of self defense, kids are much less likely to get hurt as well. You can make your point about your desire to not get bullied without all of the other unintended consequences.
Going along this route, your kids will also have a really good idea of what they are capable of, so they are more likely to stand up for themselves (like many of us tell them to do). Simply showing a willingness to fight a bully often ends any and all confrontation going forward.
What I'm getting at in all of this is, if you want your child to be able to and have the confidence to defend themselves, they need the proper toolset to do so. While I think more traditional martial arts are fantastic, they do little to help out the average student (or adult) protect themselves in a real confrontation. Knowing for sure your ability to fight with effectiveness if the need arises does wonders for ones self confidence, so much so that their need to actually defend themselves diminishes drastically.