It's funny, when I think about characters at the start of my campaigns, they are "zeroes" not because they're weak or incapable, but because they have limited influence in the world. They haven't made important connections with NPCs, found valuable treasure, or learned world-altering magic. It's not that they're in danger of being killed easily (though that can be true too), it's that they're newcomers to the setting or story.
I guess I think of "hero" in the Greek Mythology sense... Heroes are those who can do MORE than others. As characters in my campaign progress, they can do more: influence politics, raise armies, start towns, take down tyrants, attain powerful magic items...
To me, the Zero to Hero journey is all about gaining the power to influence the setting.