The method matters because the reason “for killing” is different. When a killer kills - he does so as an act of violence. When a judicial system determines a sentence of death the system and the person administering the sentence takes a human life as a punishment. If the punishment becomes an act of violence - where’s the difference between the killer and the executioner and those who sanction the acts of the executioner?
Technically, you could administer death by disembowellment, have the person drawn and quartered, boiled in oil - but by making the convict suffer through these methods of punishment, sure he’ll still be dead, but what do the people condoning the punishment in this way experience - is it justice or is it revenge - is there a satisfaction that the convict suffered?