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In Deathly Hallows, we learn that Dumbledore told Snape:
‘Tell him that on the night Lord Voldemort tried to kill him, when Lily cast her own life between them as a shield, the Killing Curse rebounded upon Lord Voldemort, and a fragment of Voldemort’s soul was blasted apart from the whole, and latched itself on to the only living soul left in that collapsing building. Part of Lord Voldemort lives inside Harry, and it is that which gives him the power of speech with snakes, and a connection with Lord Voldemort’s mind that he has never understood. And while that fragment of soul, unmissed by Voldemort, remains attached to, and protected by Harry, Lord Voldemort cannot die.’
Harry seemed to be watching the two men from one end of a long tunnel, they were so far away from him, their voices echoing strangely in his ears.
‘So the boy … the boy must die?’ asked Snape, quite calmly.
‘And Voldemort himself must do it, Severus. That is essential.’
The purpose of Harry's death is to destroy the Horcrux inside him. What reason is there to subject Harry to being killed by Voldemort? Couldn't it have been arranged more "humanely" by being done by someone else?
Here are some ideas I have for Dumbdelore's insistence, but I find none of them that convincing:
Lily's sacrifice now runs in Voldemort's blood:
Having taken Harry’s blood into himself, Voldemort is keeping alive Lily’s protective power over Harry. So Voldemort himself acts almost like a Horcrux for Harry – except that the power of Lily’s sacrifice is a positive force that not only continues to tether Harry to life, but gives Voldemort himself one last chance (Dumbledore refers to this last hope in chapter 35). Voldemort has unwittingly put a few drops of goodness back inside himself; if he had repented, he could have been healed more deeply than anyone would have supposed. But, of course, he refused to feel remorse.
Therefore, Voldemort himself needs to do the deed to ensure the sacrifice is as effective as possible
Voldemort used the Elder Wand against Harry, which can't have "fully killed" him because its allegiance was to him.
Voldemort is also using the Elder Wand - the wand that is really Harry’s. It does not work properly against its true owner; no curse Voldemort casts on Harry functions properly; neither the Cruciatus curse nor the Killing Curse. The Avada Kedavra curse, however, is so powerful that it does hurt Harry, and also succeeds in killing the part of him that is not truly him, in other words, the fragment of Voldemort’s own soul still clinging to his. The curse also disables Harry severely enough that he could have succumbed to death if he had chosen that path (again, Dumbledore says he has a choice whether or not to wake up). But Harry does decide to struggle back to consciousness, capitalises on Lily’s ‘escape route’, and pulls himself back to the realm of the living.
However a) Dumbledore wouldn't have known this when speaking to Snape (in fact, his plan was for Snape to be the the Eldar wand's owner) and b) Jo said that she doesn't see Harry's survival as a precise mathematical type of thing:
It is important to state that I always saw these kinds of magic (the very deepest life and death issues) as essentially un-scientific; in other words, there is no “Elder Wand + Lily’s Blood = Assured Survival” formula.
In "King's Cross", as well as explaining about Lily's sacrifice in Voldemort's blood, Dumbledore mentions the importance of Harry's willingness to die:
‘But you’re dead,’ said Harry.
‘Oh, yes,’ said Dumbledore matter-of-factly.
‘Then … I’m dead too?’
‘Ah,’ said Dumbledore, smiling still more broadly. ‘That is the question, isn’t it? On the whole, dear boy, I think not.’
They looked at each other, the old man still beaming.
‘Not?’ repeated Harry.
‘Not,’ said Dumbledore.
‘But …’ Harry raised his hand instinctively towards the lightning scar. It did not seem to be there. ‘But I should have died – I didn’t defend myself! I meant to let him kill me!’
‘And that,’ said Dumbledore, ‘will, I think, have made all the difference.’
Harry's willing sacrifice now protects everyone from Voldemort (just as Lily's protects Harry's), but I don't see that that has anything to do with his survival
This is just Dumbledore being himself - wanting things done "cleanly" as part of his master plan, such as making Harry destroy the Horcruxes:
Dumbledore’s betrayal was almost nothing. Of course there had been a bigger plan; Harry had simply been too foolish to see it, he realised that now. He had never questioned his own assumption that Dumbledore wanted him alive. Now he saw that his lifespan had always been determined by how long it took to eliminate all the Horcruxes. Dumbledore had passed the job of destroying them to him, and obediently he had continued to chip away at the bonds tying not only Voldemort, but himself, to life! How neat, how elegant, not to waste any more lives, but to give the dangerous task to the boy who had already been marked for slaughter, and whose death would not be a calamity, but another blow against Voldemort.
Dumbledore had a deeper meaning of the prophecy - that Voldemort has to "kill" Harry in order to be killed:
Harry understood at last that he was not supposed to survive. His job was to walk calmly into Death’s welcoming arms. Along the way, he was to dispose of Voldemort’s remaining links to life, so that when at last he flung himself across Voldemort’s path, and did not raise a wand to defend himself, the end would be clean, and the job that ought to have been done in Godric’s Hollow would be finished: neither would live, neither could survive.
Is there anything in canon that might be more convincing? May be this is all moot; as Jo says, what really saved Harry, what really counted, was something else:
What count, ultimately, are Harry and Voldemort’s own choices. They have each been given certain weapons and safeguards, but the power of these objects and past happenings lie in how they are understood, and how they are used or enacted upon. Harry has a deeper and truer understanding of the meaning of the objects and past events, but his greatest powers, those that save him, are free will, courage and moral certainty.