The monopoly on violence that defines a state is specifically the monopoly on legitimate violence; it is not a monopoly on capacity for violence but on the legitimacy of violence.
The legitimacy is a function of sufficient supremacy than any moral rightness. Otherwise organized crime and fundamentalists wouldn't be capable of reducing governments to failed state status. The legitimacy is that any who wield it openly will be crushed like if say the Hell's Angels decided to try to annex New York City by force of arms. If they couldn't then well there is your new government by strongman
> The legitimacy is a function of sufficient supremacy than any moral rightness.
It's a function of popular acceptance (or at least acquiescence) which can acheived by any combination of perceived rectitude or overwhelmingly capacity (in practice, it's usually a blend of the two, and not just one or the other, in any stable state.)