The problem is it's impossible for Tav to have a unique background that is more connected to the main story than the origin characters because the whole point of the origin characters is to play AS them for the main story. The story has to be written as if there is no Tav. So we're either left with having a background story as good as one of the origin stories, which then makes Tav essentially a customizable origin character. Honestly, I'd take that over what we have now, because right now I'm just a random person that for some reason all of these people are following around like puppy dogs even though i have no personality (head canon aside) and no background.
Something like what Dragon Age did, with a selection of background stories would be awesome, it would make you feel connected to your custom character more while also giving a more fleshed out backstory that you actually had control over.