My thoughts on this, and they are just opinions, mind you, are these.
First, we're assuming the kid wants to train martial arts.
Second, we're assuming the kid specifically wants self-defense and not sport or fitness, etc.
Third, with regard to bullies in schools...
It was always my observation that bullies work in packs. Bullies like to have their crew with them when they do their thing; they don't bully on their lonesome. Bullies are also quite often big kids compared to the others their own age. Every kid who ever bullied me was well above average in height and weight. Often a big fat kid whom even the jocks gave space to, even if the bully didn't go after them.
I don't know if grappling a school bully when they are surrounded by their howling hyenas is that great of an idea. You take the kid down (assuming you can, because he's often much larger), you submit him, and then what? You think his minions are going to let this happen without jumping in?
Personally, I recall the lessons of my father. Granted this was a different time. A bully hits you, you go grab the biggest board or stick or brick you can find and you walk up to him and lay him out flat. Busted nose, missing teeth, the whole nine yards. And you go nuts on him and make them peel you off him. Make it clear you're a dangerous SOB and have no problems crossing the line if pushed. Yes, you get suspended from school. But you also earn a safe space for yourself and all your friends; the bully will choose greener pastures and make all kinds of excuses to his pals to never bother you again.
You only have to do it once, typically. My parents moved around a lot when I was a kid, so I got to do it multiple times. The few times I decided I didn't want to fight, I got pushed into fighting anyway, after some agonizing weeks of bullying.
I think karate, TKD, etc, are fine for school kids. Yes, it can run contrary to school rules. Oh well. One of the kids in our dojo told us that his school teaches that if a bully attacks you, you should roll into a fetal position and call for a teacher. Really.
Once upon a time, I was riding my bicycle in the street in a small town, population about 400. Everybody knew everybody. I was in 5th or 6th grade of elementary school. A high school senior was hanging out with his friends at the grocery store and he threw a dirt clod at me while I was riding away. I mean he threw a fastball and it exploded in the middle of my back. Hurt like the bejabbers.
I turned around and rode back to his group. Got off my bike. They were all laughing their guts out. I walked up to the guy who threw the dirtclod and I punched him dead square in the face. I mean I had to jump two feet in the air to even REACH his face.
He looked very shocked. The laughter stopped. Then he proceeded to whup my ***. Bad.
When he stopped hitting me and I slowly got up and got back on my bike, I said to him, "You ever do that again and I'll punch you again."
He never did it again. He avoided me when he saw me. So did his friends. And all it cost me was a whuppin', and the willingness to fight back.
My 2 cents. I'd never grapple to submit a bully. I'd kick him in the slats, drop him to his knees, and then knock out his front teeth and black both his eyes. Even if I had to do it with a rock.