alt Hacker NewsNB: Weber's definition of government (which your quote misstates, so to speak, as is often the case), is that the state is that entity which has the monopoly on the legitimate claim to violence. It's legitimacy, not violence, which is key.
For the longer explanation, see: <https://news.ycombinator.com/item?id=37366751>.
That said, legitimacy is a political property, and one which cannot be attained through purely technical means. To that extent I agree with your critique.
Which itself is kind of a BS indoctrinated definition.
Wind the clock back a few hundred years and there was plenty of legitimate non-government violence. The government didn't care if the well to do dueled, the townsfolk brawled in the bar, a serial fraudster got what was coming to them, etc, etc. Sure the government could choose to care and construe it's written rules to that effect if it chose but other than exceptional situations it largely didn't and everyone alive then considered this fine.
Wind the clock back further to medieval times and it gets even messier.