"nothing is random in physics" -- That's just playing word games. If the universe is determinant, nothing is strictly random. But the collision of air molecules with each other is so nearly random that there is no hope of our detecting any non-randomness as far into the future as we can foresee.
Stars in galaxies are very much like air molecules. Their frequent individual close approaches and collisions are a matter of chance. Yet that does not preclude sharing long-term common motions at the same time, such as winds for air molecules and Galaxy rotation for galaxy stars. Birds and fish do not move randomly because they are biological organisms with functioning brains.
The orientation of the solar system with respect to inertial space is constant except to the extent that a passing star might come so close as to change it randomly. However, because the Sun revolves around the Galaxy center, the fixed orientation of the solar system in space is slowly changing with respect to the Galaxy center by about two degrees every million years.
Lots of binary stars have well-known orbits. I'm not aware of any study looking for non-randomness. But just looking at such data does not hint at any obvious pattern.
It would be an interesting hypothesis to suggest that all stars in any one gas cloud form with orientations perpendicular to the center of that cloud because supernova shock waves flatten the gas along the direction of shock wave travel. However, those stars soon separate and mix with other field stars, and one would lose all trace of the original pattern after billions of years. -|Tom|-
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