Now, I'm not sure exactly what point you're trying to make here.
Bioshock was a very linear story, and things like those radio diaries, whilst somewhat interesting, didn't do a lot to change that. They were more atmosphere, which the ME games did as well, one of the things actually better done in 2 IMO, but allowed to player to interact with some. Backstories were told through the Codex, or through news reports, a way more suited to the futuristic theme of Mass Effect. Mass Effect's story is praised thanks to its relative non-linearness. It isn't as linear as some games, yet still has you working towards an end. That end has not yet come, and will be coming in ME3, which will make things interesting.
This is, of course, coming out of the side of your argument that looks in favour of open ended stories, in which case the best type of game would be a narrated LoaP style game. LoaP being life of a peasant, a WC3 mod (I hated the LoaP titles, however there were some maps with a similar idea, that were far better executed). In such games, you could do basically anything, take any job, do any thing, and just... play. No goal, no end game. Just play. With a narrator, that is the most open ended game you can get.
The thing is, they get boring, rather fast. A game needs a goal for its story, some end for you to work towards. As such, the game must be somewhat linear. Without a somewhat linear progression, there is no agency for the player, no reason for them to keep playing.
As such, the best story in a game for you would be one that is open ended and linear in a certain balance. Something Mass Effect might end up filling once its final game is out and the whole series is taken into account as the one big story that it is.
The thing that confuses me, however, is your praise of Bioshock's story. The twist was masterful, and is among my top ten gaming moments, but the story itself isn't so great. The backstory to the world is good and interesting, but the story you act out isn't so much. The atmosphere is good, but the story is rather lacking. It is linear at its finest, and apart from the two options: Save the girls or kill them (Or save some and kill some, which resulted in the same ending as kill all but with a sad tone instead of an angry one) with two endings based entirely off these two actions. The story in Bioshock is quite the opposite of what it seems you would like - as it is incredibly linear, but it allows some exploration of the atmosphere and the area you are in.
The closest I can get to this is that you like the atmosphere of Bioshock, but not necessarily the story, and like the more open ended story types. Admittedly, the two together would be quite a nice game. Great atmosphere and great story, but I do not see what about the story in Bioshock is so great. It is linear, straightforward, and largely just an excuse to move forward and shoot things. The twist is amazing, but other than that the story is your basic 'Your trapped in a scary place, find a way out' story.