Killing someone is something you can't undo.
Even if you truly believe that someone's life is forfeit for stealing something of yours, there's no time given for evaluating the facts.
A split second decision, a bad judgement call, and you find out that the person you shot and killed, is actually some innocent bystander running to get away from the robber.
Maybe they are the person robbing you, but your shots hit someone else who's in their own house.
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.
Wow, seriously? No proportionality at all, no consideration of any mitigating factors that may be present, just simply "they broke my property rights so their life is forfeit".
I'm glad that in general society has moved on from such crude ideas of justice.
I've been mugged, and I've had my house broken into. Yes, it's shit. In neither case did I think anyone should die for it.