What was the big deal with Voldemort wanting to be the one to kill Harry Potter? Just pride, or was there something else?
He did not know about the last bit of the prophecy after all.
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The Dark Lord knew that there had been many mistakes with regard to trying to kill Harry Potter, and he knew he had made mistakes in his methods of trying to kill him. However, he was fairly sure he knew what he had done wrong, and that he knew how to fix it.
“Again, Voldemort looked up at the slowly revolving body as he went on, ‘I shall attend to the boy in person. There have been too many mistakes where Harry Potter is concerned. Some of them have been my own. That Potter lives is due more to my errors, than to his triumphs.’
The company round the table watched Voldemort apprehensively, each of them, by his or her expression, afraid that they might be blamed for Harry Potter’s continued existence. Voldemort, however, seemed to be speaking more to himself than to any of them, still addressing the unconscious body above him.
‘I have been careless, and so have been thwarted by luck and chance, those wreckers of all but the best laid plans. But I know better now. I understand those things that I did not understand before. I must be the one to kill Harry Potter, and I shall be.” - Harry Potter and the Deathly Hallows, Chapter 1 (The Dark Lord Ascending)
So, not only did he want to be the one to kill Harry Potter for the honor of it, he also wanted to make sure nothing went wrong as it had so many times before, and he trusted only himself to do it, since he was more confident in his own abilities than those of his Death Eaters. For something that had gone wrong so many times before, he trusted only himself to do it.
He also wanted to do it himself to prove that he was stronger than Harry Potter.
The curse rebounding off of Harry was the reason he lost power to begin with, and it's clear that he considered Harry as one of the biggest threats, if not the biggest threat, to his return to power, quest for immortality, and rule over the wizarding world. He also thought that his Death Eaters had begun to doubt him, and wanted to prove that he was indeed more powerful than Harry Potter.
“You see, I think, how foolish it was to suppose that this boy could ever have been stronger than me,’ said Voldemort. ‘But I want there to be no mistake in anybody’s mind. Harry Potter escaped me by a lucky chance. And I am now going to prove my power by killing him, here and now, in front of you all, when there is no Dumbledore to help him, and no mother to die for him. I will give him his chance. He will be allowed to fight, and you will be left in no doubt which of us is the stronger. Just a little longer, Nagini,’ he whispered, and the snake glided away through the grass, to where the Death Eaters stood watching. ‘Now untie him, Wormtail, and give him back his wand.” - Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)
He therefore didn't want anyone else to kill Harry Potter and deprive him of this chance to show that he was indeed a stronger and more powerful wizard than Harry Potter, which he could not do if someone else was the one who killed Harry.
Also, once the Dark Lord took Harry's blood, it stopped Lily's protection from affecting him.
The reason he used Harry's blood in the potion to restore himself to a proper physical form was to give himself immunity to Lily's sacrificial protection on Harry.
“You know, of course, that they have called this boy my downfall?’ Voldemort said softly, his red eyes upon Harry, whose scar began to burn so fiercely that he almost screamed in agony.
‘You all know that on the night I lost my powers and my body, I tried to kill him. His mother died in the attempt to save him – and unwittingly provided him with a protection I admit I had not foreseen … I could not touch the boy.’
Voldemort raised one of his long white fingers, and put it very close to Harry’s cheek. ‘His mother left upon him the traces of her sacrifice … this is old magic, I should have remembered it, I was foolish to overlook it … but no matter. I can touch him now.” - Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)
He would, of course, be the only one who had Harry's blood, which he knew then was going to counteract what had stopped his first Killing Curse from successfully killing Harry when he killed the Potters. Once he shared Harry's blood, Harry would no longer be kept safe from him by the sacrifice of his mother. While the Death Eaters wouldn't be affected by the sacrificial protection, now neither would he. So he saw had no need to outsource the killing of such an important enemy when he could have the glory of doing it himself, especially since he had more faith in his own skills than anyone else's.
The prophecy says that "neither can live while the other survives." Therefore, either Voldemort had to kill Harry, or Harry had to kill Voldemort, as happened.
"The one with the power to vanquish the Dark Lord approaches... born to those who have thrice defied him, born as the seventh month dies... and the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not... and either must die at the hand of the other for neither can live while the other survives... the one with the power to vanquish the Dark Lord will be born as the seventh month dies...."
Also, Voldemort was very prideful, and he believed that, since his Death Eaters had failed so many times, that he could kill Harry. It was not because his followers couldn't kill Harry, because Lily's sacrifice only protected Harry from Voldemort himself, as it was not a Death eater that tried to kill Harry when he was one,-
"Your mother died to save you. If there is one thing Voldemort cannot understand, it is love. He didn't realize that love as powerful as your mother's for you leaves its own mark. Not a scar, no visible sign... to have been loved so deeply, even though the person who loved us is gone, will give us some protection forever."
"But I knew too where Voldemort was weak. And so I made my decision. You would be protected by an ancient magic of which he knows, which he despises, and which he has always, therefore, underestimated — to his cost. I am speaking, of course, of the fact that your mother died to save you. She gave you a lingering protection he never expected, a protection that flows in your veins to this day."
but because Voldemort thought that he could and would be the only one able to finally kill Harry Potter.