I see this a lot and am not convinced. It appears reductionist in a way that feels like it's pushing an agenda.

Democratic governments clearly are about addressing community needs and coordinating efforts that require pooled resources (at least). I'm not denying there may be a monopoly on violence. However, in a democratic system, such a monopoly would be voted on, giving the monopoly some legitimacy (not saying it's necessarily moral).

Yet in reality, the US, for example, has the Second Amendment, which grants citizens the right to bear arms and form militias. That doesn't sound like the government has a monopoly on violence.

I guess the weasel word is "legitimate"? But is that legal or moral legitimacy (or something else)? By whose definition and arrived at how?

It feels like such a pithy comment, "a monopoly on legitimate violence", like it's expressing something deep. Yet I get the sense that supporting it requires some contortion of logic and language. Maybe I'm missing something but it doesn't seem self-evident to me at all.

I mean, the state's monopoly on violence is a legal philosophy that's been around for over a century. It isn't exactly radical or controversial.

You can start from the Wikipedia page if you're interested[0].

https://en.wikipedia.org/wiki/Monopoly_on_violence