You don't actually have to do the fight to the death, all you need is a society in which people are willing to do it. Because that provides deterrence, which eliminates most of the would-be murderers, the same as the existing system except that you don't have to deal with getting it wrong based on second hand accounts or corruption because the people handing out the consequences are the witnesses.
Notice that for the would-be murderers who aren't deterred by the threat of death, you have to do the fight to the death even under the existing system.
Anyone in any society will try to defend themselves from murder by any means available to them.
So if that's all the defense Anarchism has from murderers, it will definitely have more murderers than any society that adds any extra countermeasures.
Not necessarily. In a society where people are expected to run away and rely on the police to catch them later, more murderers may get away, because more of the victims run away instead of defending themselves but then the police never catch them.
It's obviously easier to "catch" a murderer while you're still in the room with them than after they have a ten hour head start and you have to depend on unreliable third party testimony to even identify them.
Notice that for the would-be murderers who aren't deterred by the threat of death, you have to do the fight to the death even under the existing system.