I believe the "dark time" is the present, in which the tribe is disintegrating. As Pai said, "Everybody is leaving."
Koro is looking to tribal tradition and lore to find the legendary leader who will bring the tribe back together and lead it into a prosperous future. Pai realizes that tradition has failed the tribe because there is no male heir to the leadership since her father dropped out of the tribe.
At no point in the movie does she claim to be, or want to be a leader or a special prophet. Her concept is symbolized in the scene with the outboard motor. When the rope breaks, Koro goes looking for a new rope, but Pai ties the strands together, representing the whole tribe working together.She speaks this belief in the scene at the end when she gives her speech. "We can all be leaders," she says.
Good point on the symbolism of Pai's tieing the rope and getting the motor started. Also, Koro's reaction to her successfully starting the motor shows how bitter he's become.