Imagine someone has taken something of yours- would it be just to take it back, even if it could not be proven in court? I'd argue yes, though the trick is convincing everyone else that it was just.
Imagine someone murders your brother. They are friends with the judge so they are let to walk. Have you been denied justice?
Again, the reason not to shoot people on the street is not because it is always unjust but because its not always clear what is just and we would like a fair consistent way to decide that, in a way that's legible to other people.
If you think the death penalty is fair, then you admit that you are fine with people being killed in the name of justice. The only question is, is the person the one who did the crime. Normally a judge would decide that but I don't see why that's a prerequisite