If it wasn't for the prophecy, there is a good chance Harry wouldn't have been a part of Voldemort's demise at all, though he no doubt would have fought against him. What makes him so special, so much more vital than the other students who fought Voldemort was the prophecy.
Dumbledore says something else, too, that's odd. He specifically states that Harry must die by Voldemort's hand. Why would this be true? At first, the reader might think its a requirement for the horcrux to be destroyed. However, multiple characters have destroyed horcruxes by themselves before, and the only stated requirement needed as that the object that is the horcrux must be destoryed beyond repair. Killing a human host seems like it would destroy the horcrux - as killing Nagini, Voldemort's snake, destroys Voldemort's final horcrux. So what's the difference?