Bemidbar 35:27 records that if an accidental killer leaves the city of refuge, then the relative of the person killed is allowed to kill the accidental killer. Yet, Bemidbar 35:25 records that the people are to save the accidental killer from the relative of the victim. Also, Devarim 19:1-10 stresses that the people are obligated to prepare the way to the city of the refuge and to establish even more cities of refugee if their territory becomes larger in order to stop innocent blood from being shed, i.e. to stop the relative from killing the accidental killer. This means the accidental killer is not deserving of death. Why did the Torah allow the relative of the victim to kill the accidental killer?
Luzzatto (on 35:12, see also N. Leibowitz, 1980a, pp. 189-193) explains that in olden times people believed that they had to avenge the deaths of their relatives even if the person was killed accidentally. The Torah could not change this belief immediately, and if the Torah would have forbidden the relative from killing in revenge, then he/ she would have done so in any event. The net result would have been that another person (the relative who killed) would also die for committing murder. Hence, the Torah limited the right to take revenge to outside the city of refuge with the hope that this would cause this desire to take revenge to end. The idea is that really the Torah does not want people to take revenge and this permission is based on human weaknesses. Yet, just because of human weakness, we allow a person to murder somebody who does not deserve to die? Even if taking revenge was the custom in olden times, if the relative was convicted of murder and punished, then this would eradicate this norm of taking revenge.
My guess is that the allowance of the relative to kill the accidental killer meant that relatives were a type of police to force the accidental murderer to go to the city of refuge and to stay there until the high priest died either as a punishment or as a form of atonement for the accidental killer. The permission for the relative to kill was then a cost-benefit calculation. The benefit of allowing the relative of the victim to kill the accidental killer was that this put pressure on the accidental killer to go to the city of refuge and to stay there. The cost is that if the accidental killer did not succeed in fleeing to the city of refugee or tried to leave the city of refugee and then he/ she would be killed even though he/ she did not deserve to die. The basis for the law is then that the benefit of forcing the accidental killer to flee to the city of refugee and stay in the city of refuge was greater than the chance that the accidental killer would not safely reach the city of refuge.