Yes, that will stop the script running on the client machines, so just leaving it running it on the server (thus runs ONCE). This would work perfectly as long as the "instructor" was hosting the game, but on a dedicated server, or if you want the trainees to, say, get feedback on their shots, you need to start worrying about a number of MP issues.
You are quite right in seeing that managing a single instructor that has full control and a number of simple trainees is the only reasonable way to fix it for MP (without a complete redesign). Still, this is a non-trivial alteration to make and not one I could throw together quickly.