I linked an orbit viewer above if you want to look.
Is this also random chance or is there a reason why it's so close to the plane of the solar system?
It is probably random chance, however there may be some biases from where they come from on the sky (I know people who work on that, but I don't know much about it).
N=3 does not provide very robust statistics yet, give us another decade or two.
[1] https://en.m.wikipedia.org/wiki/Extremely_Large_Telescope
https://upload.wikimedia.org/wikipedia/commons/c/c5/Comparis...
It's a fun idea thought though.
I did a full proper n-body integration and it is not visually different than this.
except that it's going the wrong way :)