I love Shadowrun, but I wouldn't want BioWare to just do the same thing. They have different styles, and romance is a selling point for many (I know, not all, but many) BioWare fans while the Shadowrun games are probably too short to spend word count on that sort of thing.
The shorter playtime I think really works in Harebrained Schemes' favor when it comes to storytelling, as it allows for a very tight narrative. They also don't have voice acting, so when you do have dialogue options, it allows for a greater variation, I think.
The Harebrained Schemes games also are staying true to the source material by keeping to a very gray tone (in terms of morality). I would think that, in the hypothetical situation that BioWare obtained the IP, they would keep to that. But that's not their IP so they do their own thing.
As far as things Shadowrun does that I think would be a good fit for a BioWare game, I think others have already said. Having stats affect gameplay in a greater way. Having multiple alternative ways to get through a situation, or have my stats affect the things I can say in dialogue, being able to talk my way out of situations - but not just via a diplomacy skill, but by having the right expertise in the right situation, etc. More non-combat related skills.
Also, just as a side note (I suppose only vaguely related now that Mr. Gaider has left BioWare...):
GAIL SIMONE @GailSimone
What is your all-time favorite pen and paper role-playing game and why? Please use hashtag @MyFavoriteRPG
@GailSimone Shadowrun, because the setting is brilliant. It inspires me as a storyteller. #MyFavoriteRPG
EDIT: I forgot to add, his name is in the credits of at least one the Harebrained Schemes Shadowrun games as a backer, I think. I remember seeing it there.