There are a few things that trouble me with this post.
That's one...Disney takes safety VERY seriously. I don't know if BTM has it, but a lot of rides have safety interlocks, so that if something like a lap bar isn't in place, the system won't allow it to leave the platform and the CMs have to check it. If it doesn't have it, I thought they DID have to physically check.
Yes. This. Unless something has changed, I distinctly remember some rides being slightly delayed, and then a CM walking down and checking all the lap bars...usually stopping at one to fix it. On more than one occasion too.
Fourth, the OP says the headmaster had all six kids write her letters of apology - for that, they would have had to have known exactly which six kids it was - either it wasn't a large group so they were easily identified, or they already knew they were a problem, perhaps because Disney DID take action already, just not known to the OP.
I didn't even think about this, but you are exactly right. How would they know exactly who it was - after the fact.
I could be wrong about this, and it could have happened as the OP says...it just seems highly, highly unusual.