Murderers could run to one of the cities of refuge and be protected by law from retribution. However, I don't understand how the guy desiring revenge is any less a sinner.
It wasn't a case of desiring revenge. It was a duty which had to be performed.
Núm 35:15 These six cities shall be a refuge for the sons of Israel, and for the stranger, and for anyone who stays among them, so that everyone who kills any person through error may flee there.
Núm 35:16 And if he strikes him with an instrument of iron, so that he dies, he is a murderer. The murderer shall surely be put to death.
Núm 35:17 And if he strikes him by throwing a stone, with which he may die, and if he dies, he is a murderer. The murderer shall surely be put to death.
Núm 35:18 Or if he strikes him with a hand weapon of wood, with which he may die, and if he dies, he is a murderer. The murderer shall surely be put to death.
Núm 35:19 The revenger of blood himself shall kill the murderer. When he meets him, he shall kill him.
Núm 35:20 And if he stabs him from hatred, or hurls at him by lying in wait, so that he dies,
Núm 35:21 or in hatred strikes him with his hand so that he dies, he who struck him shall surely be put to death. He is a murderer. The revenger of blood will kill the murderer when he meets him.