And what means do you think society uses to protect the rights it decides to grant or deny? If a trans person shows up to a doctors office with a gun and demands healthcare, is that how we know their rights are protected?

You just jumped from me talking about society using violence to talking about an individual using violence. In this current form of society in the US, individuals are not endowed with the authority to use violence to demand goods and services, only society at large is, via legislative means. So if society says (via a law), "your office must provide this person healthcare without discriminating against them" and the doctor ignores it, and they are criminally charged, a warrant will be issued and a policeman with a gun will be sent to arrest them, etc etc.

We know their rights are secured when they can walk in public freely without harassment and go to the doctors freely and legally to get the care they need.

Yes, because if someone harasses them, they are protected by the society assuring those rights (with violence, or the threat thereof).

You’re falling entirely into a political frame that benefits the far right libertarian perspective.

Not at all. You just think I am because you are conflating me stating that violence underpins systems of authority with me being pro-violence. I'm in fact very much against violence, and against systems that do rely on it, like non-consensus based democracies and other systems which assert authority over people unwillingly based on their geographic location.

Violence is antathema to the democratic process...

If you are literally talking about violence as part of the process itself, that is obviously not part of our system (unless you break one of our laws about voting, in which case violence enters the room to arrest/ punish you).

and [anathema to] a truly peaceful, cooperative society that is based on the rule of law.

I agree with you up to "society", because yes, violence is obviously exclusive by definition of peace. But the second half, about the "rule of law", just shows a fundamental misunderstanding of those words, and their relation to violence. "Rule" in that phrase literally means "the exercise of authority or control over". So the Law's exercise of authority or control over (members of a society). That is, in a non-consensus-based system, enforced with the threat or actualization of violence against the members of that society. ALL authority is backed by violence. That's the problem with authority.

Being able to live a peaceful life without fear of being murdered or lynched is a fundemental right.

It obviously should be, I agree. Often that's not the case, due to either violent individuals, or society's authoritative laws being used against people to hurt them (like Florida, Texas, and many others are doing to trans people and many other minority groups right now).

Not some kind of “other” category that someone should expect to have to carry a gun around to ensure. Just like when right-wingers carry around guns, it’s an illusion of control and safety, not safety rooted in the society they live in.

Once again, you are conflating the people in the article talking about protecting themselves from individuals and non-authority entities, not about ensuring their rights within the framework of society at large. No one is setting up an LGBT+ defense ranch in e.g. Florida to provide protection against a state government.

And I agree that people should not HAVE to carry around a gun to be safe; that's the whole ostensible (but false) promise that our society extends to its citizens. But our society doesn't actually operate like that, because it, like all other modern nation-states, was founded through violence (revolutionary and settler-colonial), is enforced and maintained through violence (police and military), and in our case exports violence around the globe.

If you are hoping for the US to ever be a country that does not exist in a perpetual state of violence at all levels, I think you are naive.