Indeed, letting individuals decide who does and doesn't deserve to live, is how we end up with endless blood feuds. "They killed my brother so their life is forfeit."

I've read a saying that a government is an entity with a "monopoly on violence." I think that however imperfect it is, giving the government this monopoly means that there's an objective process for deciding who deserves to live, and who doesn't. Personal satisfaction as a criterion is always going to produce outcomes that are massively disproportionate, and will simply shift the dissatisfaction to someone else.