I didn't watch this anime, but isn't the whole premise that Light Yagami can ONLY kill them if he knows their name?
So if their name is already known...they are probably already arrested

Pro:

1. More "ethical" form of the death penalty
2. Reduce prison fees

Cons:

1. Completely disregard the prison system
2. Justice is in the hands of a dictator
3. Nobody has any hope of reform

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The social consequence would mean nothing. The only way this would impact anything is if Light Yagami was the new name of the death penalty. I'm not sure if his deaths are painless, but if they are, they might be the new standard for the death penalty and that would be "moral" in today's society. If he is rogue, he would fall into the cons more.