That's not how randomness works I don't think. For instance the most random sequence of numbers someone can choose for the lottery is 1,2,3,4,5,6.
I would argue that the fact that they are close to the same plane is a sign that they are perfectly random. I think you are assuming that an appearance of a pattern equates to non-randomness.
I read a Chinese astronomy book a few years ago that addresses the issue I think you are facing. Western culture has equated the stars (the heavens) with perfection and mathematics. So when something isn't perfect, or cannot have a math formula attached to it, they dismiss the underlying issue, and keep perfection. This was when the Chinese discovered sunspots, yet western intellectuals dismissed the idea of spots on the Sun, because it meant the Sun wasn't perfect.
As well, it also had a huge impact on the acceptance of the geocentric model of the solar system. The epicycle model invented by Brahe only had two adjustable parameters and was considered mathematical perfection, but was in truth, partly false. The objects did not orbit the Earth.
I want to say this nicely because it means a lot to me, but I think you have adopted the idea of perfection in the stars, when in fact, chaos rules. I've learned that over the years. Mathematicians and astronomers/astrophysicists do not like randomness, it makes them uncomfortable. I think you should read, "The Drunkard's Walk: How Randomness Rules Our Lives".