Why did the Central Tibetan Administration get involved? It's easy, they want to control everything, even the Dalai Lama if they can. That's why now there's 2 Panchen Lama, 2 Karmapa, and many double Tulkus or Rinpoches out there due to CTA recognising their own candidates for control. What's the difference between them and what they claim China did then - to install their own China Panchen Lama?
To a certain degree, the CTA is controlling the Dalai Lama. The Dalai Lama was used by his closest people to make money. He is 'sold' like a merchandise. CTA and the Dalai Lama's representative arranged a public talk and appearance of the Dalai Lama with a cult leader. They have received about 2 million dollars for the arrangement.
The Dalai Lama is a monk, he does not need the money. As a monk, it is his duty to spread Dharma, there should not be money involved. CTA and the Dalai Lama's closest people are very selfish, they only think about the money but never take into consideration the reputation of the Dalai Lama. The Dalai Lama worked very hard to bring the Dharma to people around the world and his hard work is damaged by these greedy people.
The CTA has no intention to maintain harmony and unity in the Tibetan community, this is why the Free Tibet movement is a failure. Since the Karmapa issue is not going to bring any benefit to them, the CTA will also be quiet about it. It has been so long since the dispute started, nothing has been done by the CTA as usual.