Headmaster was not so much a title or a job description as it was an identity. There were so many identities that Albus Dumbledore had occupied in his long life. Son. Brother. Friend. Lover. Enemy. Professor. Chief Warlock. Supreme Mugwump. Savior. Mentor. Of all these identities, Headmaster would be the one that lived the longest, through the minds of his students. To the many witches and wizards who passed through the doors of the Great Hall for the first time to be sorted and saw Headmaster Dumbledore sitting in the center of the head table, he would be frozen in their minds like that forever. Old, wise and a bit batty - Headmaster Dumbledore.
To a handful of those students through the years, there was more. Some saw him as a challenge - students like the Marauders and the Weasley Twins tried to get reactions out of him and make him proud. Students like Lily Evans and Hermione Granger wanted to impress him with their ability to work hard and outperform their peers. It was only a very few students who ever saw Albus Dumbledore the man.
Tom Riddle was one such student. Full of hubris himself, he strode into every classroom as though he owned it. He saw past Albus the Professor and saw a man with strengths and weaknesses to be admired and feared.
Harry Potter was another student who saw him as a man. Dumbledore had tried to remain a mystery to this student more than most, but in the end it was to this student he was laid completely bare as a man of mistakes and victories. Harry had little use for Dumbledore's titles and legendary status, being in possession of many of his own. Seeing past all the layers was second nature to Harry. Harry possessed a natural ability to do the opposite of whatever you'd expect another student to do only rivaled by his friend Luna Lovegood. It was that essential nature that made Dumbledore confident that he would wear his legend status better than Dumbledore himself.
There was only one student Dumbledore had ever seen himself in so strongly. Draco Malfoy arrived at Hogwarts confident that his way was the only way view the world. Draco possessed an ambition to do great things and see his name written in the history books that rivaled the best among Slytherin House. He loved his family more than he loved their ideals. Dumbledore saw something of the boy he had once been in Draco Malfoy and it made him reflect on a life of what if's and could have been's.
What if someone had helped steer him off his path earlier on? What if he had been able to persuade Gellert follow him towards a different future? What could have happened to them if they had learned the error in their ways as boys instead of fighting over them as men? For all that Albus Dumbledore had become the one to defeat Gellert, he still wished he had been the one to save him instead.
Dumbledore was a lot of things, but Headmaster would be what he was remembered as the longest. He couldn't save the Savior, but maybe he could save Draco Malfoy. Maybe this boy who had so much potential and so much pressure on him could be shown a different way and a different path. If he was bound to direct Harry towards his doom in order to save everyone else, could he not at least find one soul to protect himself?
So it was that Albus Dumbledore committed himself to a path of giving Harry every tool he needed to defeat Voldemort and try to cheat Death, and simultaneously pray that Draco would make it to the end with a soul worth saving.
Draco wouldn't remember the night on the Astronomy Tower as the night that Dumbledore achieved his last great task. He'd remember it as a nightmare, a horrific episode of his young life where he came close to murdering someone in cold blood. That was the night he'd remember as a turning point in his life, realizing that no matter how much he loved his mother he wasn't the man he'd been raised to be. He wasn't a killer. This realization was the achievement Dumbledore had prayed for, and even if short lived he knew it was his from the look in Draco's eyes. A look he'd long wished he could have seen in Gellert's eyes, a look that spoke volumes about what the future wouldn't hold.
Slowly influencing events around Draco Malfoy's education to show him the essential value of each human life was his magnum opus. People might accuse him later of manipulating Harry, but they didn't see the way he truly tried his best to manipulate Draco into seeing the error of his father's beliefs. Perhaps a man even greater than him could have resisted the temptation to do it, would have left Draco Malfoy's future up to fate.
If he could have left Draco something in his will explaining all of this without placing him in greater danger, he would have. As it was, Draco was left with the only gift Dumbledore would ever give him. Compassion.
Harry's power was love, but Draco's was compassion and in the end it took both these powers to defeat Voldemort. Without Draco's compassion for his classmates they could have died that night in Malfoy Manor. Without Draco's compassion Harry wouldn't have had the chance to overpower him and become the owner of the Elder Wand. Without Draco's compassion perhaps Crabbe or Goyle may have just killed Harry outright in the Room of Requirement. Without Draco's compassion at Malfoy Manor perhaps Harry would have left him there for the FiendFyre to consume. Without Draco's compassion perhaps his wand wouldn't have worked quite so well for Harry.
That night on the Astronomy Tower Draco learned that he was more like his mother than his father, and that he was perhaps more like his estranged Aunt Andromeda than he had ever cared to consider. He didn't believe that blood status made any difference in magical ability or anything else. The realization felt like a curse, but Albus Dumbledore cared little for what the realization would feel like to Draco as long as he had it.
In the moments that it took Severus to utter the killing curse the Headmaster felt the relief that he had finally prevented at least one tragedy from ever happening.