And I think a key question would be “why did you agree to do this role when it wasn’t in your contract?” - and I think legally I would be seen as the responsible adult (despite what the organisation tell me).
That would be my concern too.
When it comes to things at work and being asked to take on responsibilities that aren't mine, my question to myself is always, "If the worst happens, could I justify my involvement to the authorities/investigating body?" If the answer is no, I just say I won't do it without appropriate safeguards for all (including me) in place.
If I accepted a role knowing it compromises me or someone else, then I'd consider myself complicit in the failing.
I wouldn't accept a verbal assurance either.
In your case, you feel uncomfortable with it and now have a reason to be (the broken bones) so I'd ask for a meeting to clarify your responsibility around it because you feel it even if they are saying its not yours. Where is your protection?
If they insist, I'd ask for clarity in writing so that your role and level of responsibility for it is clear along with who does ultimately.have responsibility for it.
Hard to say without knowing the exact details but appreciate you won't want to put them here necessarily.