Rashi does note that the Kohen ha'gadon should have prayed that there be no accidental murders. But I'm wondering if anyone has heard of a more direct relationship between the life of the murderer and death of the kohen.
(What vaguely comes to mind is the passing of Aharon ha'Kohen (ha'gadol) in the beginning of Parashat Masei... perhaps his love for life, for others, and his potential willingness to sacrifice his life for others. But not sure what about the essence of kohanim and murderers.)
The Rambam writes that the death of the High Priest distracts the mourners from seeking vengeance on the possible killer in the city of refuge:
The chance of returning from the exile depends on the death of [the high-priest], the most honoured of men, and the friend of all Israel. By his death the relative of the slain person becomes reconciled; for it is a natural phenomenon that we find consolation in our misfortune when the same misfortune or a greater one has befallen another person. Amongst us no death causes more grief than that of the high-priest. [Guide for the Perplexed 3:40]