"Secondary" does work in place of "minor":
coming after, less important than, or resulting from someone or something else that is primary
So the primary/secondary pair can work for major/minor but I don't see anything wrong with using major/minor. The pairings are synonymous.
That being said, I find major/minor to be more common for things like sports leagues and injuries:
He plays in the minor leagues
We treat major injuries before minor injuries
I do consider primary/secondary more appropriate for locations:
This is our primary site.
Take this to the secondary office
My hunches are saying that this is partly politics. "Secondary" sounds less diminutive than "minor" or more of a shift in focus than of size. The idea that the secondary and primary items could swap positions is more becoming than swapping the uses major and minor.
In other words, major and minor tend to apply better for objects of different scale or size. Primary and secondary are more apt for diplomatic or focal designations.
Another point worth considering is the addition of "tertiary" to the primary/secondary pair. I am unaware of a suitable third option for major/minor.
As for you proving your point, you probably will not be able to without building a better consensus. The major/minor pair does work in this situation. If you really wanted to go the extra mile you would want to check other uses of major/minor and primary/secondary for applications to locations.