The state defined by its monopoly on violence. Period.

No, the state has a monopoly on initiating violence. All citizens have the right to violence if it’s in self defense.

Some states will even let you carry firearms around in case you need to apply violence in self defense.

That's nitpicking. It is clear that failed states are those with nontrivial scale violence perpetually initiated by non-state actors. The strength of the state equates to its capacity to limit the scale of others' violence. US gun laws, globally speaking, are a statistical anomaly, and if you started using guns against others with any scale or frequency the government would surely intervene.

None are more hopelessly enslaved than those who falsely believe they are free. - Goethe

US gun laws are irrelevant, since the legality of using violence in self-defense (note that we aren't even talking about lethal force here!) is applicable to a lot more than guns. And yes, in most countries, it is legal, although certain limits of proportionality etc may be imposed, and there may be a duty to retreat if possible.

But also, US gun laws insofar as they pertain to the use of guns for self-defense aren't as anomalous as many people think they are. There are a few other countries in which it's possible for a regular citizen to get a permit to carry a handgun openly or concealed - Czechia, for example.

It’s not nitpicking. Almost all states recognize the legitimate use of violence for self defense, even if the specific methods permitted vary on a state by state basis. Thus if your definition of a failed state is one in which the government doesn’t have sole right to violence, then you end up with the rather absurd position that all states are failed.

When generalizing it is taken for granted that one discusses the general case not an extreme case. It is not possible to hold a discussion in general terms with people who insist on nitpicking. All generalisations are invalid. Some are more invalid than others.

When engaging in discussion it's taken for granted that you agree with your own opinion.

It's useful to have additional conversational tools to engage positively, such as:

'oh yes, that is an improvement on what I said', or

'I appreciate what you said but I was trying to make a different point, which is:', or

'That's a useful way of capturing what I was trying to say', or even

'Yes, I see that what I said was not quite right, thank you'.