> Their primary purpose is to prevent lethal intergroup violence and blood feuds.
A state's primary purpose is to maintain a territorial monopoly on legitimate violence and to secure its continued existence. Generally this is at odds with its founding myth, but it's pretty obvious that states are also subject to a process that resembles evolution by natural selection. That said, I'm not even aware of any notable state or state-like entity that was created with the goal of "ending blood feuds". The vast majority of these territorial monopolies were created by means of conquest where the goal was to acquire resources owned by the people in some territory or with the goal of pooling resources to defend against such conquest.
> maintain a territorial monopoly on legitimate violence
For what purpose? Seems strange you start here and then need me to connect the dot to blood feuds.
We are saying the same thing. If adjoining families take it upon themselves to engage in blood feuds this is in direct conflict with the state's "monopoly on legitimate violence".
The point of the phrase is to convey that the only legitimate violence in the conception of the state is the state's own violence, or in some select cases (self-defense being one) violence the state explicitly grants or endorses. The monopoly on violence is thus sort of required when deriving the moral theory. The states violence is by definition legitimate in its own eyes. The monopoly is the operative part - there can be no competing violence. This is perfectly coherent, whereas a competing "legitimate" violence would not compute given the internal logic of the state itself. It would be existential.
This is the basic germ of politics. Prevent small conflicts within a group from escalating and prevent external domination. I mentioned both. Again, not sure where you see disagreement between our views. Preventing conquest by a roving band is compatible with my view. If the conquering entity is a state then my contention would be the process of stopping blood feuds would have already occured (among the conquering people) as that is what law and order is, which is sort of a prerequisite to effective armies. You don't get to just kill the person who killed your brother absent state intervention, because such acts have a way of spinning out of control. Justice is the province of the state alone. Of course, blood feuds still happen. Occasionally they are even legitimized (as is necessarily the case with this sort of intervention) by the state. But as state power increases, the space for this sort of violence shrinks. Once again, tacking on "legitimate" to "monopoly of violence" often confuses the point. The point is the state requires that all violence it does not sanction is necessarily illegitimate. There is a nice circular logic here, evocative of the phrase "the house always wins".
Put another way - when the state does it, it's not illegal. Those who control the state control the law.