Harry precisava entender por que era imperativo que ele mesmo morresse para destruir completamente Voldemort e garantir a morte de Voldemort. Aquele pequeno pedaço de alma não tinha que viver como tinha sido (hesito em dizer "morra" aqui porque foi o último pedaço da alma de Voldemort que se tornou a criatura mutilada e parecida com um bebê em King's Cross - não estava vivo, nem morto, mas no limbo.).
Por que ele teve que ser dito especificamente que a alma de Voldemort se apegou a única coisa viva na sala, infelizmente canon não responde a essa pergunta. Talvez J.K. Rowling tem, no entanto. Se eu puder encontrar uma citação, vou editá-la.
ETA: É importante lembrar que Harry NÃO é um Horcrux, apesar do que Dumbledore diz.
JKR diz se Harry é ou não um Horcrux adequado:
"Here is the thing: for convenience, I had Dumbledore say to Harry, "You were the Horcrux he never meant to make," but I think, by definition, a Horcrux has to be made intentionally. So because Voldemort never went through the grotesque process that I imagine creates a Horcrux with Harry, (SU: Mm-hm.) it was just that he had destabilized his soul so much that it split when he was hit by the backfiring curse. And so this part of it flies off, and attaches to the only living thing in the room. A part of it flees in the very-close-to-death limbo state that Voldemort then goes on and exists in. I suppose it's very close to being a Horcrux, but Harry did not become an evil object. He didn't have curses upon him that the other Horcruxes had. He himself was not contaminated by carrying this bit of parasitic soul."
[SNIP]
"I do think that the strict definition of "Horcrux," once I write the [Harry Potter Encyclopedia], will have to be given, and that the definition will be that a receptacle is prepared by Dark Magic to become the receptacle of a fragmented piece of soul, and that that piece of soul was deliberately detached from the master soul to act as a future safeguard, or anchor, to life, and a safeguard against death." JK Rowling - Pottercast 12.23.07 - The Leaky Cauldron
Então, sim, o chapéu detectou a porção da alma de Voldemort dentro de Harry. Mas não, Harry não era uma Horcrux oficial.