I think you're wrong about how the other nurses on your unit, and other people generally, would react to the idea of 'heroic responsibility', depending on you were to both bring it up and present it.

The key part of the quote with which I would expect lots of people to agree is:

“You can’t think as if just following the rules means you’ve done your duty."

I'd expect everyone to have encountered an incompetent or ineffective authority figure. I'd also expect nurses to routinely help each other out, and help their patients, by taking actions that aren't formally or technically their responsibility. Ex. "Did Ms. Smith ever get that pillow she requested?"

But fully accepting heroic responsibility means you're also accepting responsibility for (a) doing what needs to be done despite feelings of guilt; (b) not burning oneself out (unless that's the best course of action); (c) not accepting heroic responsibility for one thing (unless that's all you truly care about).

Accepting heroic responsibility for all your patients very well might be best honored by you doing whatever you need to do "to care less – and thus be less frustrated and more emotionally available to comfort a guy who was having the worst week of his life".