The best argument for the death penalty is about legitimacy of the government. Given the choice between revenge killings and the death penalty sentence in fair courts, we should choose the death penalty. Of course, this presumes fair trials, etc., so if the trials themselves are non-existent or unfair, the calculus changes. And if the impulse for violent revenge does not exist among a people, then the calculus changes as well.
To be clear, the argument isn't that repaying murder with death is just. The argument is that to maintain a stable government, the government needs to respect popular opinion on the issue.
All that being said, I don't have any particular insight into Erdogan's changes. But changing rules just after a traumatic event can lead to regrets and overreach, as the U.S. found post 9/11.