No, no, that’s not what it means here at all. Maybe the dictionary is referring to another use of まま, but it’s not this one. In fact, I wrote about it on this forum a while back in one of our analysis threads. I did an extremely etymological analysis of まま. In essence, it means ‘state’ or ‘condition’, and sometimes refers to a sort of ‘status quo’. So yes, @Shannon-8, you remembered correctly.
I think the 2nd is closer to ‘Why didn’t you let/make him die?’ (Of course, ‘let’ makes a lot more sense here.) The difference between the 2nd and the 3rd is that 〜てやる is like 〜てあげる, so it has the nuance of doing something for someone else, and makes it clear that the speaker considers letting the old man pass on without having to undergo a pointless operation a good thing, almost like it’s a ‘favour’ they could have done for the old man.
EDIT: BTW everyone, so… gonna be needing to spend more time studying stuff that isn’t Japanese. I’m actually in prépa, (I think YanagiPablo knows what that is. Zizka might as well.) which is normally quite intense. I need to revise coz the entrance exams are happening in about a month, it’s just that I haven’t been in the mood to study lately. I might still come online during breaks, but knowing that I’m kinda addicted to giving detailed explanations and so on, I need to limit my time here (so I don’t keep reading new threads and writing new explanations, and so I don’t eat up time for actual breaks). Continue to feel free to tag me, but also know that I might not come online at all if you don’t (except in the evenings, French time). When all is said and done, hopefully I’ll be admitted to a good engineering school in August, and I’ll be able to restart my Japanese studies in full force, but for now, that’s that.