It’s about divorcing doing the right thing from social rewards and status–the former does not automatically lead to the latter, and achieving the latter doesn’t mean you’ve accomplished the former. Sansa was brought up equating the two, because the songs and stories she loved told her that’s how it worked. If you save the day, you get the crown and the beautiful girl, and if you’re, say, an attractive blond prince, you must be a good person. She learns better by the end of the first book.
The triumph comes from facing a situation in which you will not be rewarded for doing the right thing–it might even get you killed–but you do it anyway, to emphasize that heroism means something more than being the main character and ending up on top at the end.
“No chance, and no choice” from Brienne VII AFFC is probably the best example in ASOIAF. Brienne is heavily outnumbered by Rorge’s wrecking crew and knows she is almost certainly screwed if she steps out from the shadows and challenges them. She could very easily hide, let them inflict horrors on the kids at the inn, and move on safe and sound when they’re gone. But despite having “no chance,” she takes them on anyway, because in order to be the true knight she wants to be despite everyone telling her she can’t, she has “no choice” but to defend those kids to the last breath. After all: “I charge you to be brave, I charge you to be just, I charge you to defend the young and innocent.” Like her ancestor Dunk, she’s “a knight who remembered his vows” while technically having never taken those vows–again, this is the author separating the values of knighthood from the social systems in which they’re embedded. Gregor may have been anointed by the crown prince himself, but “he was no true knight” where it counts.
See also Waymar Royce’s brave last stand against the Others in the series’ opening pages, demonstrating there’s more to him than bluster and arrogance, or Davos doggedly sticking to his mission in White Harbor even though the reasons to turn back keep piling up, or Stannis telling Jon that he intends to let the wildlings through the Wall despite the certainty that this will hurt his prospects with the Northern nobility because “when the cold winds rise, we shall live or die together.” There’s an immense catharsis to be found in these moments, in which characters go through intense crucibles and discover what their values really are when the pretense of being rewarded for them is stripped away.
Excellent!
To add one point, it’s not just “divorcing doing the right thing from social rewards and status,” it’s also about choosing to do the right thing even if you might not succeed in accomplishing your goal. Hence why we append “existential” from Existentialist philosophy – a school of thought that argued that the universe lacks inherent meaning and that the individual creates meaning in spite of it – to the term.
Thus(to introduce a couple more examples), Syrio Forel wins an existential triumph because he makes the decision to remain true to his beliefs (what Camus called perseverance), even if means his own death; indeed, even if Arya had been captured anyway. Just as Yoren’s last stand in defense of his charges and the values of the Night’s Watch is not diminshed by the fact that he fails, Syrio’s triumph is not dependent on the practical outcome of his actions.