IMO it wasn't really so much about the effects on the story as it was giving the player a chance to have a greater personal involvement in different parts of the game, i.e., playing as the Human Noble made the stakes higher when you were dealing with Howe since he killed your parents and usurped your family's title, the Dwarven Noble had more of an involvement in the succession conflict in Orzammar (and was personally fucked over by it), the Mage had a personal conflict with the Circle, etc., etc. I'll grant that this was imbalanced; the Dwarven Commoner origin has fuck-all to do with most of the goings-on in Orzammar and instead you get a minor link to the crime boss there, for example, and for the most part these links are superficial and don't give you any unique options in dealing with their respective linked parts of the story.
All that to be said that I wouldn't mind something similar in Cyberpunk - a little playable intro section for each background that shows how you get caught up in the events of the game, just to give you a little more grounding in the world right off the bat.