EDIT: Thorin did not have the ring of power, it was lost with his father.
Tolkien wanted to rewrite the Hobbit, but since he wasn’t able to, Peter Jackson brings to the screen the significance of Thorin being the last of the royal line of Durin, along with Fili and Kili, based on Tolkien’s appendices.
In LOTR, darkness has fallen over the world. Sauron has grown strong. Why?
Because the bearers of the rings of power are all dead, or under his control (except for the elves). Rings, plural.
“Three Rings for the Elven-kings under the sky,Seven for the Dwarf-lords in their halls of stone,Nine for Mortal Men doomed to die,One for the Dark Lord on his dark throneIn the Land of Mordor where the Shadows lie.One Ring to rule them all, One Ring to find them,One Ring to bring them all and in the darkness bind themIn the Land of Mordor where the Shadows lie.”
So of the men, none of the rings are left. The kings have been turned into the 9 wraiths.
The dwarf kings? All dead. All their lines lost.
Gandalf is so insistent Thorin take back his homeland because it is a dwarven stronghold. It is a kingdom from which a great force of good (ie the dwarves) can defend against Sauron.
Thorin lives in Ered Luin with his people. Probably in nice homes, or decent ones. But if Ered Luin was attacked, he could not defend it. He would need a stronghold. Erebor.
Sauron is desperate to kill Thorin, and he has to deal with this on top of having gold sickness (which may even be caused by the ring), on top of the Arkenstone being generally seductive as hell (even Bilbo was affected by its beauty!)
Thorin had CORRUPTING FORCES (Sauron, arkenstone, gold sickness) bearing down on him in the middle of a quest to kill a dragon and then having to fight off armies of orcs while not trusting the elves who came with their own armies and the men being angry at him.
Sauron was desperate to kill Thorin. He was desperate to destroy the last stronghold of the dwarves, and desperate to ensure that the line of Durin be destroyed. Thorin’s madness didn’t come down on him just like that—it must have been years in the making. Thorin wasn’t weak willed or dark-hearted, but he was being corrupted on all sides, and Sauron wanted him dead.
He did the same to Aragorn. He was desperate to kill off the line of Elendil, the Numenorians, which was why Aragorn had to live in secret all his life.
Sauron needed Thorin dead. Which is why Thorin is so important, and why his death would rule the fate of those who came after him—in the Lord of the Rings.
If Thorin doesn’t die… then Sauron is defeated (easier, at least) and the world would be saved.