As you point out - and eli-sennesh points out, and the trope that most closely resembles the concept points out - 'heroic responsibility' assumes that everyone other than the heroes cannot be trusted to do their jobs.
This would only be true if the hero has infinite resources, actually able to redo everyone's work. In practice, deciding how your resources should be allocated requires a reasonably accurate estimate of how likely everyone is to do their job well. Swimmer963 shouldn't insist on farming her own wheat for her bread (like she would if she didn't trust the supply chain), not because she doesn't have (heroic) responsibility to make sure she stays alive to help patients, but because that very responsibility means she shouldn't waste her time and effort on unfounded paranoia to the detriment of everyone.
The main thing about heroic responsibility is that you don't say "you should have gotten it right". Instead you can only say "I was wrong to trust you this much": it's your failure, and whether it's a failure of the person you trusted really doesn't matter for the ethics of the thing.
My referent for 'heroic responsibility' was HPMoR, in which Harry doesn't trust anyone to do a competent job - not even someone like McGonagall, whose intelligence, rationality, and good intentions he had firsthand knowledge of on literally their second meeting. I don't know the full context, but unless McGonagall had her brain surgically removed sometime between Chapter 6 and Chapter 75, he could actually tell her everything that he knew that gave him reason to be concerned about the continued good behavior of the bullies in question, and then tell her if...
[Originally posted to my personal blog, reposted here with edits.]
Introduction
Something Impossible
The Well-Functioning Gear
Recursive Heroic Responsibility
Heroic responsibility for average humans under average conditions
I can predict at least one thing that people will say in the comments, because I've heard it hundreds of times–that Swimmer963 is a clear example of someone who should leave nursing, take the meta-level responsibility, and do something higher impact for the usual. Because she's smart. Because she's rational. Whatever.
Fine. This post isn't about me. Whether I like it or not, the concept of heroic responsibility is now a part of my value system, and I probably am going to leave nursing.
But what about the other nurses on my unit, the ones who are competent and motivated and curious and really care? Would familiarity with the concept of heroic responsibility help or hinder them in their work? Honestly, I predict that they would feel alienated, that they would assume I held a low opinion of them (which I don't, and I really don't want them to think that I do), and that they would flinch away and go back to the things that they were doing anyway, the role where they were comfortable–or that, if they did accept it, it would cause them to burn out. So as a consequentialist, I'm not going to tell them.
And yeah, that bothers me. Because I'm not a special snowflake. Because I want to live in a world where rationality helps everyone. Because I feel like the reason they would react that was isn't because of anything about them as people, or because heroic responsibility is a bad thing, but because I'm not able to communicate to them what I mean. Maybe stupid reasons. Still bothers me.