Collective wage and benefit bargaining tends to be the main focus of many unions. I'd say in the instance of this particular Nurses Union the union also advocated well and often for the safety of their nurses. So yes they do ensure worker safety but I don't think it's thought of as much as collective bargaining.
Worker deaths have halved since workplace safety came into play in 1970, this wouldn't have been enacted if left up to companies as it is much cheaper to find an uninjured worker than it is to implement safe working conditions (I'm generalising obviously but I think it holds). It would have been heavy collective bargaining from unions that made this happen.
Also thank you for sharing the original story it was a really interesting anecdote.