The rest of your argument is a straw man. Did I make a gun control argument? Did I say you should be incapable of self defense? I did not. Anticipating this exact stupid argument, I made it clear that I (a) appreciated self defense training and (b) have my kids involved in it.
Responding to a schoolyard bully with a punch to the gut is one and the same as responding to a B&E with a shotgun. If you want your kids to do one but not the other, you're sending mixed messages. You don't respond to barroom bullies with a punch to the gut--you get the bouncer's attention. But that's tactical judgment. If you're in the schoolyard, getting a teacher's attention will just as often do nothing, and if it accomplishes anything it only makes the problem worse. It's more like a prison, where the wardens will allow all kinds of abuses happen without interfering.
"Don't throw a punch, ever" is pacifist crap, not tactical judgment.
In adult life, you have the legal right to physically confront burglars and drunken assholes on the street. It takes a degree of tactical judgment and skill to know how and when to confront them, but outright pacifism is wrong for two reasons:
1. It leaves you vitally incapable of handling emergencies. If someone breaks into my house, brandishing a shotgun isn't going to solve the problem completely, but it is going to ensure my immediate safety until the police arrive.
2. All you're doing is delegating the physical violence to someone else, which means "implicitly trust authority". In the school system, the authorities don't care. In real life, the authorities are the bullies more often than not.