You don't understand bullies or bullying.
The vast majority of bullies only understand the threat of force. They don't give a damn about detention, suspension, admonition, or anything that 'authority' figures will do to them. Those are merely inconveniences that reinforce their perception that their target was too weak to deal directly with them, and instead went to the 'authority'. Ninety-nine times out of one hundred, the bully will ramp up their harassment of their target in order to 'prove' that they are unaffected by any such attempts to mollify them. They will then double their efforts to intimidate their target into accepting their subordinate position.
This is a 'victory' for the bully.
I know damn well that this is exactly how it works having been the victim of bullies from the age of 6 to 16.
Here's what works: You HIT them. You HUMILIATE them. You INTIMIDATE them with a greater show of force, and if that means you have big, dangerous friends, so be it.
Unfortunately, we are still animals and many of us behave that way. Bullies are no different. They are the low ranks in a civilized society, and as such are less affected by the ministrations of civilized 'authority'. Therefore, if you want to shut a bully down, as I learned to do, you have to lower yourself to their level and be uncivilized. This doesn't mean assault, it means anything that will show them that their behavior as an animal will put them in a place that actually frightens them.
NOTHING a school can do will intimidate a bully because, as I said, physical force and intimidation are the world they like to live in. Once you show them how vulnerable they can be in that world, they will start to immediately see the benefits of civility.
You'd be amazed at how immediately well-behaved and civil bullies can become once they've had the right incentive. I've seen it first-hand every single time I've put the above into practice.
So when a linebacker threatens a bully, you can bet the farm the message is received loud and clear and better behavior will ensue.