Love this post.
Sectarianism wasn't something that Buddha taught. No where in the Kagyur is there anything on sectarianism. Therefore it could only arise due to the ignorance of man.
Buddhism has one root - Buddha. Whatever school you learn from and practise with, it leads you to the same ultimate goal. The path may differ but the achievement/goal is the same. There is no difference in the goals of each school.
It is like maths. the fundamental concepts of mathematics is the same. You can become a mathematician from doing your studies at cambridge, oxford, imperial, australia, canada. Whatever school you choose, the end result is that you will become a mathematician. Same thing. There is no division in the schools.
Sectarianism arose from schismatic mindsets.
Karmapa and Trijang Rinpoche are best of friends, because their fundamental goal is the same. To benefit others and lead others out of samsara.
To add on this, it is common and well known also that in every major religion there are sectarianism. In Christianity, we have Catholics and Protestants. And within both, we have many different schools as well sometime 4 to 5 sub divisions. In Islam, we have Sunni, Shiite and Ismailis. Hindus have different focus on the deities and the same with Taoism. So what is common here is the human factor. Human are the culprit. The Buddhas and the great saints know this and tries to cater for this in their teachings. However, the human karma are such that no matter what, there are conflicts between these different schools.
So, we call this sectarianism. It involves disparaging one and another, focusing on differences and "faults" of others. The Buddha taught the Dharma so that we can discern the falsity behind all these "samsara" delusions. We need to practice the Dharma and be more discerning and find the similarity in these different schools. After all, all the teachings depending on the scope is to find happiness and lessen sufferings for beings. It is humans who causes these suffering.
Certainly not the Lamas. For what they appear to do are mere reflection of our though, be it wisdom or delusions.