Allegorically, the relative who seeks to avenge the victim’s blood is our own evil inclination. It attempts to trick us into sinning, thereby causing us to suffer some form of spiritual “death,” i.e., a loss of vitality in our spiritual life. The Messianic Redemption will be our ultimate refuge from this pursuer, for the evil inclination will be nullified in the Messianic future. Similarly, the future resumption of the Temple service will afford all who need it the opportunity to complete their atonement.
In the meantime, the study of the Torah is our refuge from our evil inclination, for the holiness of the Torah has the power to neutralize the effect of evil on us.1


I emphatically disagree with the statement that a person who accidentally kills someone is a murderer (as in the example of the axe head flying off and striking someone), as there is no malice. Further, I disagree even more with the view that the accidental killer has some sort of a "defect" that can only be rectified by his scurrying off to a city of refuge.
Let's be more mindful of our interpretation of scripture and our use of language in same. Shalom.