If Jacob kills David by accident, then the blood can be avenged by David’s relatives. Jacob’s only way to stay alive is to run to a city of refuge. And there he stays until the death of the current Kohen Gadol.
The linkage is very difficult. Why would an act of unintentional manslaughter be linked to the Kohen Gadol in the first place?
Whenever the Torah or halacha refers to what the Kohen Gadol does, it refers to the Kohen (whoever he may be) as “Aharon” – who was, of course the first Kohen Gadol, and the archetype for all of his successors.
But Aharon was also involved in an accidental killing of his own. When Aharon helped to create the golden calf, he had no intention of having the situation get out of hand, and eventually lead to the death of 3,000 people. It was manslaughter writ large.
And yet, Aharon was never punished. G-d never mentions it again, and neither does anyone else. Even in his last days, when G-d says that Aharon cannot go to Israel, he does not mention the episode with the golden calf.
Perhaps we can think of it this way: When Aharon died, the manslaughter died with him, unavenged by G-d or the surviving relatives of those 3,000.
So when the Kohen gadol dies, the relatives can recall that if G-d (and the Jews in the wilderness) could forgive Aharon for his accidental homicides, then they should also be able to forgive the killer. The justification to avenge a killing dies when the descendant of Aharon does.