As I've stated, "not all killings are murders" - murder constitutes an unjustified killing. Wars are tricky, but if one participates in a war for all the wrong reasons - and kills others in an offensive manner based on deceit and lies - then it boils down to murder. However, if one refuses to fight in a war, but is dragged into war by being attacked while being in a peaceful state - then defending oneself and one's loved ones by killing others in respond cannot be seen as murder.
In addition, yes, mistakes are made in terms of giving innocent people a death sentence, but should we ban all cars because of accidents on the road? I think not. And a life sentence places a burden on society. Just imagine, I murder someone, which constitutes a death sentence on that person without due process, and then society has to provide accommodation, feed me, etc. It is certainly not fair. If one takes another person's life in an unjust manner, then that act should automatically cancels one's right to life.
Now of course, in instances where one is not dealing with a hard-core murderer, the person confesses and show true remorse - hard labor and a long prison sentence could help toward rehabilitation. If the accused is too weak and/or not willing to work, then he/she must face capital punishment. It sounds harsh, but you want something that serves as a strong deterrent.
On the other hand, if I am accused of murder and receives capital punishment - and it came out later that I was innocent, then those who falsely accused me must face capital punishment themselves. Justice must be served either way.
The problem today is that murderers have more rights than their victims. It is absolutely insane.
With your statement about capital punishment being falsely accused and then giving capital punishment to those who falsely accused you is that the best approach? Give this scenario you were falsely accused and executed on the belief that you committed the crime. New evidence comes up that the person who accused you did it falsely. So let's execute them. Oh wait new evidence appears to have debunked that the person falsely accused you.
There we have it. No justice what so ever. Everyone involved in the case has now been executed. Only their families are left to grieve. This actually happens quite regular in minor cases (not murder) although it's not completely bonkers to assume that this could well happen and may of happened in the past.