So after thinking about it some more, I came up with a possible rationale/rationalization why a wizard's death might be needed.

Assume the "script kiddy magic" theory is right - A powerful wizard can be bind complex magic into a simple to execute script, with a key phrase (and/or emotion or gesture). Thus it wasn't some perverse law of the universe that decided "Wingardium Leviosa" is how levitation is activated, but some perverse ancient wizard.

A Horcrux stores an image of you, and the activation sequence is bound to the death of a wizard. It was meant to be an emergency backup script, activated on the death of the wizard. I.e. the ancient who created it was thinking that when a wizard dies, they would automatically be backed up into a Horcrux. This explains where ghosts come from, and why the ghosts we know of were all wizards. Later, someone figured out how to activate the script without dying. Unfortunately, the method they discovered involved killing another wizard.

A backup is limited by the hardware that runs it, so ghosts, which can only barely be said to run, don't seem like real people. They have limited ability to form new memories, so they seem more like chatbots than people (in the MoR universe). A Horcrux is even more limited unless it can get near a brain, but has some "upload" magic associated which means it can possess people under the right circumstances. Harry could be a Horcrux, in the sense that he might contain a backup image of Voldemort, but it can't (normally) run for much the same reason Voldemort can't cast a spell on Harry. That's why the hat didn't sense it.