We're already given two choices for the player: either a heavily scripted Origin character or a blank slate for roleplaying.
A middle option with some kind of "light" personal quest would be hard to pull off without steering too far in one of those two directions. Too much scripting and it would feel like your story is on rails, not much different from choosing an Origin character. Too little would just be a "meh" side quest with no real impact. That's without even getting into how it would have to fit each race and class.
It's an interesting idea, I just don't think it's practical, considering the opportunity cost in developer time that could go somewhere else.
Not sure how giving you personal quest puts you on rails. I mean... not anymore than having the tadpole puts you on rails. There IS a plot. So making part of that plot specific to you is not restrictive.
Specific to which me? I have a Wood Elf Thief, a Tiefling Warlock, a Drow Thief, and a Drow Ranger, off the top of my head, which toon are we writing this for? What happens when more races and classes are added after EA ends and the game launches? How much time should they dedicate to that now, instead of working on whatever's missing from the next two chapters, and all the bug squashing and polish that needs to be done? Time is a finite resource, and it has to be managed well if they're going to get the game out "on time". I would just as soon they spent that time between now and release fixing what needs to be fixed, and getting the rest of the game ready for launch, than hand holding me through who my custom character is.