Tom
Basically speaking the teacher is responsible while the kids are under their care and oversight, including areas outside of the school(building) e.g. during an excursion, as long as the kids acted in a foreseeable manner (the younger the kids the broader the meaning of "foreseeable"). This appears to apply here.
Assuming the teacher didn't (sternly) warn the leading kids to stay away from the stairs, and make sure they actually did, and continued to make sure they did, they were probably neglecting their duty. I would think the "didn't see it happen" alone is enough in an ordinary situation to demonstrate neglect, especially if the allowed area included the top of the stairs.
That said, the teacher can't be held responsible, the Gemeinde is. This is explicitly regulated in the (law on liability) Haftungsgesetz , see §4. Whether the Canton has taken out 3rd party liability insurance can't be any of the damaged person's concern, at all.
ETA:
Also see this leaflet by Luzern . Especially:
"Pupils are under the care of the teacher during school time. During this time, the teacher must take all reasonable precautions to protect the life and health of the pupils entrusted to him or her. Therefore, a teacher can also be prosecuted if a learner does not actively injure himself but injures himself by omitting a necessary action of the teacher [this is the verbal translation] due to an action the teacher omitted: e.g. reconnaissance of a hike, clarifying learner skills, lack of instructions to the learner, etc."
"Schülerinnen und Schüler befinden sich während der Schulzeit in der Obhut der Lehrperson. In dieser Zeit muss die Lehrperson alles Zumutbare vorkehren, um das Leben und die Gesundheit der ihr anvertrauten Lernenden zu schützen. Deshalb kann eine Lehrperson auch dann strafrechtlich verfolgt werden, wenn sich ein Lernender zwar nicht aktiv, sondern wegen Unterlassen einer notwendigen Handlung der Lehrperson verletzt: z. B. Rekognoszieren einer Wanderung, Fähigkeiten der Lernenden abklären, mangelnde Anweisungen an die Lernenden usw. "