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Just like Slughorn, Albus Dumbledore collects people. Only, instead of focusing on those with influence, he looks to the outcasts.
The expelled half-giant. The young werewolf. The repentant Death Eater.
He protects them and gives them a second chance. All he asks in return is their loyalty.
And, if on occasion he requests that they undertake a certain task, invoking their debt of gratitude - well, that is no more than he is owed.
He once thought to add a certain disowned Black to his collection, but quickly realised his mistake.
Sirius is not an outcast, but a rebel. He knowingly chose his path, and chooses what price he is willing to pay for it. He refuses to be used.
So Albus Dumbledore abandons him.