That's sort of the point. Each playthrough is its won story. This is always going to be true as long as we have any input into the game at all; allowing us to break BioWare's story is just one extra step.
I'm perfectly aware of the kind of storytelling you want, but it's not what everyone wants, nor is it "just one extra step." It'd have to be a complex compromise between various design imperatives.
The game might play very differently with no crew (some would say badly, but that's a value judgment I'm not willing to make), but as long as the consequences flow from your actions, that's a good thing.
Of course consequences would be easier if we had far fewer people examining our actions, but that's not what we want. We don't want to just input our actions into a box and get a number of different cutscenes as a result, we want frequent feedback, preferably tailored to our specific actions.
I'm fairly confident in saying that a BioWare game without companions would be almost assuredly worse than one with companions. Given how critical characters are to BioWare stories, it may even be plain bad.
I'm not sure what you mean by "fumble up most of a rescue mission".
The doom scenario is where we mess up the whole mission, i.e. a plot-critical character dies so we get a failure state or a "doomed story" indicator. The doom-1 scenario is where we save the plot-critical character(s), but everyone else dies. How will these people act differently if one of the nonessential characters is a family member? What if we could save two of this essential NPCs children, but only one survives? If we're given full control of our actions, what happens if we save the essential NPCs, but deliberately kill the others?
This design would have worked fairly well in DAO, KotOR, ME1, and ME3. Each of the major story quests stood alone. ME2 could have handled critical mission failure much the same way it did the failure to do loyalty missions (reducing chances of ultimate success).
The design would work fine (though scaling it beyond 50-ish of hours would be extremely difficult), but complete modularity isn't what I want. I want characters stories that interlock and react to each other, and I don't want the individual quests to seem suspended in time until I arrive. I'm perfectly willing to relinquish some of my roleplaying freedom to have that.
BioWare's stories always involve some villain taking action, and the player character responding to that action. I suggest that the line of quests that make up the core of the story should have failure conditions along the way without breaking the story - just changing the nature of the final conflict or the ultimate outcome. And if they did, those quests could even be skipped, thus invoking the failure event by default.
What quests are necessary? How far do they need to be progressed? What does the final conflict look like if I don't complete certain quests? How will the game achieve a smooth sense of progression if I'm popping in and out of various individual missions?
This is especially true if ME:A is another trilogy (or just maintains continuity like DA). What happens in the next game if you don't begin a certain plot thread? What happens if you failed it? I'm particularly attached to the notion of save importing, and I do like the idea of more branching stand-alone games, but I do think the continuity of BioWare's IPs gives them a great degree of appeal. I don't think BioWare will ever go back to single-game story arcs with Mass Effect (or Dragon Age for that matter).