We claim to have understood much about our galaxy without going beyond even our solar system - which is just a small portion in one of the arms of the galaxy. I assume that the plane in which planets move around the sun (the Eliptic) is also in the same plane in which all other arms of our galaxy rotate about its's center. Would our understanding about our galaxy different, if the two ecliptics would have intersected each other at some major angle?
You can go outside tomorrow morning just before dawn (written 02/05/22), look at the spectacular array of planets (Venus, Jupiter, Saturn Mars) on show at the moment and confirm for yourself that they do not all lie in front of the Milky Way. Here is a star map (for London, tomorrow morning at about 04:30) making the point.
The array of planets defines the plane fo the Solar System (a handy ecliptic line is also shown in the picture). The plane of our Galaxy is marked by the Milky Way - also clearly visible (although possibly tricky just before dawn).
The point is, these planes do not line up at all. We think that the plane of the Solar System is randomly oriented with respect to the Galaxy. There have been no hints of systematic alignments for other planetary systems either. What we understand from this is that the process of star and planet formation takes place in turbulent environments within giant molecular clouds. The turbulence scrambles the angular momentum vectors of the collapsing clumps such that no alignment is expected.