Everyone always wonders about this, like it's some kind of big mystery. But Dumbledore doesn't 'choose' to kill Grindelwald, not in a cold, premeditated sense. That whole 'Greater Good' philosophy they dreamed up in their youth? It's the thing that chains them together until the very end. Dumbledore ends up having to stop it, to dismantle their shared legacy of arrogance. He's spent decades haunted by Ariana's death, unsure if it was his curse or Gellert's that killed her. Going to Nurmengard isn't about a vendetta; it's a penance. He has to be the one to end it because he's the only one who ever truly understood the scale of their mistake. The duel is less about killing and more about a final, terrible accounting.
Plus, let's be real, Grindelwald by 1945 is a genocidal monster holding Europe in terror. Dumbledore, for all his later faults as a manipulator, is the only wizard alive who can take him down. Not killing him would be an act of incredible moral negligence. The choice is between letting a tyrant continue or doing the ugly, necessary thing. Dumbledore's tragedy is that he's uniquely qualified for both roles—the only one who loved him, and the only one strong enough to end him.
He walks away with the Elder Wand, the last relic of their broken dream, and that feels like the real punishment.