On the contrary, I would like to propose the possibility that it was not at all necessary for Voldemort to kill Harry in order for the piece of Voldemort's soul to die; however, Dumbledore always has his reasons. Dumbledore wanted Voldemort to be the one to "kill" Harry because Dumbledore had, as usual, excellent foresight, and because Voldemort fully believed in the prophecy.
Recall that Voldemort himself insists that he must be the one to kill Harry Potter:
"There have been too many mistakes where Harry Potter is concerned ... I understand those things that I did not understand before. I must be
the one to kill Harry Potter, and I shall be." — Voldemort in Deathly Hallows, Chapter 1
This is the not the first we hear of Voldemort being reluctant for anyone to kill Harry but him; shortly after Dumbledore's death, Snape stops the Carrows and Rowle from killing Harry:
"Have you forgotten our orders? Potter belongs to the Dark Lord — we
are to leave him! Go! Go!" — Snape in Half-Blood Prince, Chapter 28
Obviously Voldemort doesn't want anyone else killing Harry (and Dumbledore ought to know this, via Snape). He wants to do it himself, because he does not think anyone else would be able to get it right, especially because his plans to kill Harry have been foiled so many times. Only by killing Harry himself will Voldemort be able to truly ascertain that he has succeeded.
In addition, Voldemort has heard bits of the prophecy, particularly the bit saying that "either must die at the hand of the other." He's had some time since the Ministry debacle to mull over what he knows about the prophecy, and he must have come to the conclusion that, as long as Voldemort is alive, nobody will be able to kill Harry Potter but Voldemort himself.
That explains Voldemort's insistence that Harry must die at his own hands. So why did Dumbledore insist the same?
Consider what would have happened if, during the end of the seventh book, somebody else had inadvertently killed that piece of Voldemort's soul within Harry, with Harry being sent into that same King's Cross limbo state. Voldemort would have been immediately suspicious, because he had not seen to it himself and because that would be a direct contradiction of the prophecy. He would have gotten down to the bottom of it and realized that Harry was not actually dead. He may even have realized that Harry really had developed some sort of Killing Curse immunity; he may have retreated and looked for ways to get around it. Even if, ultimately, he would have been unsuccessful, who knows how much longer the war would have been, how many more innocent lives would have been lost in the process?
Dumbledore would have wanted the war to end quickly. He knew that Harry's sustained blood protection was a powerful weapon that could bring about a decisive victory if used correctly and at the right moment. They had the element of surprise on their side, so why not keep it that way?
If Voldemort himself is the one to cast the Killing Curse at Harry Potter, it will be Voldemort himself that will be utterly bewildered that Harry was still alive: he's followed the prophecy and he's seen it through with his own eyes and wand. Dumbledore may have banked on the fact that Voldemort's arrogance, ignorance, and shock in the heat of the moment would cause him to act rashly upon failing to kill Harry. Instead of rationally assessing the situation, Voldemort stupidly does the same thing yet again and expects a different result, which led to his demise.
I can't say exactly how much of this Dumbledore anticipated, but in any case, it was clearly advantageous from a strategic standpoint for Voldemort to have been the one to "kill" Harry and not anybody else.