I think origin characters will always have a better story and connection to the world because it's handcrafted. However, I think it would be cool to have a character quest based on your class or background of your custom character, so you could feel more of a part of the world. It would be hard to implement without it being really vague, why does my trickery cleric get the same quest as my tempest cleric? I guess they could do subclass, but that becomes a crazy amount of content to make.
