jane, your idea of removing the bullies is an interesting idea, but as you note in your post the admin often doesn't know who the bullies vs. the bullied are. They literally don't know. My oldest was in a class that was having trouble at recess and lunch with bullies, and two kids were routinely being ostracized and if they sat at the lunch table others would jump up and move away. So the teacher devised a seating chart that she thought would work by placing like behaving kids with each other and intended to enforce that chart in the cafeteria until dynamics calmed down. I was class mom that year and got a call from one of the bullied boy's moms - kind of hysterical - saying did you even look at the chart? My son has been placed sandwiched between the two biggest bullies!! I couldn't believe it either, the teacher literally had no idea who the ring leaders were because they were very slick and socially savvy. I talked to the teacher and we re-arranged the chart that would work better - and I was left kind of amazed that the teacher honestly couldn't tell who the ring leaders were in the 3rd grade. I wouldn't call her unusual, either, as you state - teachers can't always see it and the victims are seen as instigators.
So I don't know how you'd come up with a beneficial way of removing the bullies - often it would be further that the victims would be isolated for appearing to be the problem.
And before anyone says it's easy for teachers and admin to spot bullies - it isn't. It's easy for parents to spot them somehow, but it's harder for school admin to see it.