I'm opposed to KSP having a story. The wonderful thing is that every time I play it, there's a different story, and things can play out differently. I can start at the dawn of the space age, or as a small start-up, or anything I want. Your story is going to be different from mine, which is different from his, etc.
If every time I played KSP, I got the same storyline, the same missions, the same "accomplish this task before you can do something else", I would quickly get bored and the replay value would drop to zero. Having a more "open" story, there's more freedom, but the same series of events and goals would also reduce replayability. But I would still be stuck in someone else's vision of KSP, playing out what the writer thinks it should be.
Adding on a new mode might work, if I am able to ignore it and continue playing how I want. Allowing users to create a series of missions, setting the starting conditions, a series of goals, scripted events, and a persistent world between them would be a good idea. This would allow those who wish to write a story in KSP to do so, and it would provide other players with a large supply of different storylines to explore, which helps keep replayability on the table.