Right now in Venezuela theft on the street (burglary, pick-pocketing, shoplifting, etc) has become a de-facto capital crime because thieves who get caught are spontaneously set upon by bystanders and killed on the spot.
This societal change was not centrally ordered or organized - it arose spontaneously and instinctively once the resource situation became sufficiently dire that shoplifting became tantamount to attempted murder. It also used to be a capital crime to steal horses in the American West, for the same reason (survival without your horse was unlikely).
So it seems to me that capital punishment is an instinct produced by natural selection that acts as a societal immune system. A society that abandons it completely is immunocompromised and won't survive very long.
Absolutely the decision making process for enacting it should be as error-free as possible, which is the traditional argument for leaving it to the government (a theoretically neutral, objective third party).
Of course the government can also be corrupted and introduce error in the process in the form of false positives and false negatives, but that process can only continue until the situation becomes intolerable to the point that the population spontaneously reclaims that authority as they have in Venezuela.