Legalise murder, rape and robberyWatch
Disclaimer: This in philosophy, that means I'm trying to instigate philosophical discussion of the concept not outright condoning it.
You would be reducing investigative accuracy, imbalancing the 'justice' the criminals would face, and also legalising revenge. I think it would be reasonable to view this proposal as encouraging rather than discouraging murder, rape and robbery.
Show me a town in history that has been raped and pillaged, but nobody died.
I'm not saying it's the right way to go about it, but history is on the side for the OP's argument.
Anyone who died as a result of persons intending to rape or pillage would have been murdered or manslaughtered. I agree that legalising rape and robbery would most likely increase the murder rate, but it would still be the murder rate. And a cull implies some criteria to warrant death - culls aren't wanton.
I think that avengers have historically not been particularly dissuaded from enacting their revenge from the threat of further revenge. Revenge also tends to escalate violence; it is rarely as clinical as an eye for an eye.