You mean just like Dragon Age Origin has done it? Or how your background came up in Mass Effect and Starfield?
This is not an impossible task. Other games already do it. Larian chose not to make the background matter. Like they chose to give everyone the same background no matter how little sense it made and abandoning the background system halfway during development like so many other things.
I haven't actually played all these games, so correct me if I'm wrong, but itsn't Mass effect PC an Origin character ? And in dragon age your class and race determine the origin, so the choice is much more limited, right ? Not saying one is "better", but different players have different preferences. No one wants all games to be the same. I'm happy with a totally free custom build and I don't mind making up my own backstory.
In Mass Effect you can chose several backgrounds for Shepard which each leads to some changed text and a unique quest during the game.
Starfield gives entire questlines and flavor events based in your background like your parents being interactable NPCs, who also barge into your place at work once and annoy your co workers. And Dragoon Age gives each race/background combination a unique prolog chapter and changed dialogue and sometimes options in the main game. And except for Mass Effect none of those were Origin characters.
