Anarchism

I think that others have done a sufficient job of answering the main question. I have things to say about the secondary questions:

how badly damaged is it?

I am wondering how it would be possible to reverse or remove that opinion of so many.

I've actually put quite a bit of thought into this problem. Countering the intentional destruction of language in general seems to be basically impossible to do directly because of how many people there are who have no particular interest in figuring out what the word really means, and who just perpetuate the current zeitgeist via sheer inertia. It also doesn't help that there are a great many who claim to be anarchists who actually want the term to be misunderstood. The nihilistic version of anarchism that you're calling out is perfect for sociopathic individuals who want a world without accountability.

My conclusion is that the word itself has become effectively destroyed and unusable (except in contexts where you know that your audience is made of the small subset of people who actually understand what it is supposed to mean) and needs to be replaced. However, attempting to just invent a new term for the same ideas won't quite work either, because it will just be equated to the old one, and destroyed by the very same actors as before. We may be able to buy some time, but we need to do something about the forces that work to destroy the language itself if we want a lasting solution.

We need to learn how to protect ourselves from the actors that consistently sabotage our efforts to communicate, form communities and institutions, and actually accomplish objectives. We need to learn how to recognize those actors reliably, and keep them out of our spaces. Feds and such aren't really the main issue here - it's the people that claim to be our allies but instead subvert our rhetoric and activity to their own selfish ends that we need to be most wary of.

Once we can keep our spaces clean, we'll have control over our language again, and we can use a new term or the old one. Those who are actually interested in doing good can be kept safe from the interference of bad-faith actors as long as they are able to find their way into these spaces.