It's been a while since I looked into the exact reasoning, and I'm not sure that Erik Curren's book Buddha's Not Smiling: Uncovering Corruption at the Heart of Tibetan Buddhism does a goot job at summarizing it either, hence, he wrote a book. The answer, just like my post the other day regarding the death of the Sixth Dalai Lama, involved a strong background and basis in the sects of Tibetan Buddhism, the internal politics regarding the Shamarpa Lama, the Tai Situ, the history of the previous Karmapas, the life of the 16th Karmapa, the politics of 18th Century Tibet, and the politics of Tibet-in-Exile. And while I could explain each of those elements in turn, it's probably better for me to direct you Curren's book where he ties all of those relevant elements together, rather than me blindly accusing the Tai Situ or the Shamarpa of forgery and falsely claiming tulku properties for their own religious, political, and financial control.

That said, Curren's book is by no means objective and he states so from the beginning, making it clear that he has a clear preference for candidates of vying for the Black Crown. When he discusses the arrival of the two candidates, he speaks about them in totally contrasting tones, the first being presented as a black horse candidate who was falsely propped up, implying that there was some bribery involved. And the other presented as appearing gloriously by some call from the gods. (I'm paraphrasing, but I think anyone who read the book will agree with me about the tone problem.) Another relevant passage, IMO, tastelessly discusses the candidates' genitals in relation to the age they were supposedly claiming (i.e. implying one of them was born too early, prior to the 16th Karmapa's death). Maybe it's just me and a holdover from my puritanical upbringing, but it left a bad taste in my mouth.

That said, Curren certainly knows what he's talking about, even if he is biased. (And at least he knows he's biased.) He's well acquainted with Tibetan history and literature, and I think his documentary analysis of the letter presented by the Tai Situ I think is on point. Not only that, but so far he's the best we have. There are plenty of books out there that discuss the lineage of the Karmapas in various detail and present the 17th as their preferred candidate. Of course, these are religious lineage histories, not historical texts, which I can say that Curren's, despite the text's flaws, does a good job of outlining and describing the circumstances that led to the current state of a split lineage. And I would say that while he leans towards one side, he leaves room that someone who wanted to do more research could easily fall on the other side, or just take a non-sectarian approach, as both Karmapas generally seem to take (they've mentioned repeatedly their desire to meet, though for all we know, it might be like going back and time and meeting yourself, destroying the timeline, so maybe it's for the good of the universe that they don't? I certainly can't think of a specific, single incident where two rival claimants to a tulku lineage actually met...). Or as the relatively famous (here in Europe at least) Nydahl's have taken, wanting to stay above the political bickering and maintain a non-sectarian approach to the spread of the Dharma.

And really, I think that's Curren's goal, no matter how far he leans. He sees the Dharma and its spread as being hindered with sectarian in fighting, so he wanted to write a book to highlight what happens when ambitious and greedy individuals seize the reigns of religious sectarianism for their own purposes.

Anyway, I hope that helps and let me know if you have trouble finding a copy of the book!