Ni instead finds the new perspective that explains why process X isn't working right.
Ne starts with kind of a home base that might be regarded as Si (I don't particularly mean the function, but rather as the other end of the Ne/Si dichotomy). There is an internal subjective point of view, "the box" if you will, and it works outward from that. Notice that the notion of "house" and "apartment" and "rent" and all the rest remained constant. You put the concepts together in a new way.
Here's an example of Ni from my perspective, of the "spooky" sort. I was once watching one of those corny Twilight-Zone-like shows back in the 90s [Tales from the Crypt, if I recall correctly], where there is always a twist in the plot, usually totally unexpected. This particular story was about an old man who has died, his pretty trophy wife, and his two sons. There is also another son who left long ago, estranged, and is only mentioned in passing. They're arguing about the inheritance, and the sons don't want the trophy wife to have a share. Without going over the entire story, which I barely remember, I'll tell you my Ni insight. I thought, "Oh, the wife is really the missing son, with a sex change," after watching it for about 5 minutes.
I was proven absolutely correct at the end of the show!
The "context shift" part is obvious: I replace "wife" with "son," which seems absolutely ludicrous, absurd.
What went on inside my head is how the context shifts happen. I'm given elements A, B, C, D, and formula f(A,B,C,D), which is unknown. f(A,B,C,D) is the context. I switch from f() being the story in its context to the context of "what would I want to do to give this story a twilight zone flavor with elements A, B, C and D?" Remember, one of the rules is that they have to tell you everything you need to know, so that they can point back at it and show how you were looking at it all wrong. At this point, the missing son was highlighted as a problem element in the story. He was mentioned too much. In a typical story, he might show up in the finale, and reconcile with everyone, including the trophy wife. But that option I rejected as too normal. The most "elegant" solution was that the missing son was already present, and the only way for the missing son to be already present was for him to be the trophy wife.
The most important aspect of Ni in this regard is that we don't disregard particular possibilities based conventional notions of likelihood, but rather we allow/disallow contexts based on whether they "work." As long as the context is self consistent (kind of like Ti), we'll keep an open mind about it. If it's the ONLY possible context, it sounds like we just predicted something magically, by "just knowing." It even feels like that to ourselves.
In the more conventional arena, I use Ni to troubleshoot. The context shifting in this regard is to come up with a set of possible problems that could have happened, no matter how ridiculous they might seem. I investigate the most likely ones, and quickly find the real answer, which is often but not always my first guess.
In one particularly odd case, a web page was crashing based on some weird SQL error. There was no way that anything was wrong. All the data looked correct. Everyone was puzzled. So I looked at the data for oddities. The main weird thing I saw was that the person's name as given in the data was very long, basically a sophisticated identifier for QA testing to sort results. I changed the name to "Joe Smith" and the bug went away.
At that point, I knew that something about the name (it turned out to be the length) was somehow corrupting the data. It took forever, though, to explain this to everyone else. Their reaction was always, "No way," and "That makes no sense at all." But I could point at empirical data to prove it, which isn't often the case for Ni. In spite of the empirical data, the conclusion was so odd, that it wasn't easy for others to absorb. My Ni attitude was of the "I don't know why it is true, but I know that it -is- true" and I knew that I would figure out why eventually, and didn't need to know "why" to communicate the problem. It turns out that Microsoft's SQLXml had a bug in it, and we needed to update to a new version of SQL Server to fix it.
This is how Ni relates to Se. The "singular vision" is often an Se-perspective of the matter, either we want to make the Se-perspective true (by understanding and controlling our environment via Ni) or we want to understand why the "Se-fact" is the way it is. This is entirely analogous to how Ne branches off of an Si-subjective understanding.
I hope these anecdotes give others a good understanding of Ni. The results are weird, but all we're doing is admitting possibilities that others immediately discard, because they don't fit those others' context(s).