Taking something I wrote earlier and expanding upon it a bit...
I actually rather like the proposition that Shallan's father "gave" her the sword.
The reasoning goes a bit like this: Shallan is a major viewpoint character for the whole series - book 2 or 3 will have her as the main focus (Brandon hasn't decided yet - read the Q&A links from his latest blog post), which would include full flashbacks, presumably to particular scenes only hinted at so far. I wouldn't be surprised if those flashbacks were already written, since a LOT of back-story was written already apparently. So, I wouldn't be least bit surprised if there are some really good answers to the things we're discussing here. Answers that make us go "argh, now why didn't I think of that earlier!?".
If Shallan's father "gives" her the sword he could also give her a reason not to sell it at the same time. ie some kind of dying message - in a similar way a certain king gives Szeth a particular gem to keep. Effectively, one thing can neatly solve two problems - why did Shallan of all people get the sword, and why didn't she sell it to save the house?
I'm certainly not saying that this is the only solution, but rather that the solution might be something along these lines.
One thing the book makes very clear is that Shallan finds conflict disturbing and always has and angry people make her very nervous. Her brothers tease her for being shy. This seems likely due to her father - after all, he was so bad he seems to have "broken" all of his children in some way, and that Shallan's 3 tutors left shortly after starting because of his temper. It's only more recent that Shallan has started becoming bolder. It's a bit hard to square this with Shallan killing her father somehow, but clearly it did happen - or at least, Shallan thinks she killed her father (can that spren tell the difference?). Personality-wise and culturally-wise, Shallan is the last person you'd expect to kill her father and also the last person to get a Shardblade. She'd need a good reason for picking it up if she did it on her own initiative. It's very hard to think of a good reason for her to pick it up and keep it if her father died and it simply appeared. She might be curious enough in general but if her father had just died I don't think she'd just be that interested at the time.
Also, Brandon doesn't seem to do "just so" stories that much. There's clearly a lot more to her father than meets the eye - more than can be explained with just flashbacks. If Shallan's father gives her the sword (or wills it to her) then it makes him and his backstory a lot more interesting. It would also make Shallan more conflicted about what she feels about him, which is right up Brandon's street.