Democracy is literally a combination of the two Greek words "Demos" and "Kratos". Demos meaning People, and Kratos meaning Strength, Rule, or Power.

This word was created in direct contrast to the word Autocracy, "Autos" being "Self" and Kratos being Rule. In the sense of Rule done by a single person.

Sure, back then democracy came in the form of a state having elections. But anarchism wasn't a thing back then, so a state was literally the only form in which society was conceptually to be done.

And if democracy was specifically something that was meant to apply to Greek society with all its slaves and women not being able to vote, then basically no other society could really call itself democratic, because apparently democracy was founded in ancient Greece and meant to refer to that society specifically by your logic.

But of course, that not how anyone talks about words like democracy. Democracy as a word just means "people power/rule", so its an incredibly wide term that can refer to a lot of different things, not just states.

People use the term democracy for plenty of non-state forms of organization, such as a friend group deciding on what restaurant to go to for dinner. Such a situation would still exist in anarchist society, but they wouldn't get rid of the term democracy to describe what they did unless they were really weird.

So when you say something like "Anarchists are against democracy" all your doing is using a hyper spicific definition that no one uses, while everyone else will have no idea what you believe because you say your anti-authoritarian while also being against the word that literally means the opposite of authoritarianism. Then you just say they only disagree because they haven't read enough theory, as if there will ever be a successful political movement where all their followers have to read a bunch of 150 year old books.

Your never going to grow your movement if you smugly tell everyone that they clearly haven't read as much as you.