Instead of "I'm anxious," try "I feel threatened"

While I find your solution thought provoking, I am not sure I agree with your conclusions. Suffering from chronic anxiety myself, I know it not to be a cognitive process. My problem is not that I spend effort cogitating on a solution to a non-existent threat, but rather that the constant feeling of dread poisons my everiday existence. False attributions are a typical byproduct of this state: one rarely experiences such feelings without also being compelled to find an explanation as to why they came about, arguably, because in a healthy mind, one would expect that such states of mind would only appear when warranted by an external event. However, the problem with chronic anxiety is that there are no such events. The creation of false narratives prompted by a natural desire for explanation only serves to maintain or magnify the state. In the examples you provided, cases where your solution worked for you, the state of anxiety is in fact prompted by real events, no matter how insignificant they may appear to be, and I would argue that the reaction to them was a normal flight or fight response, albeit one that was maybe exaggerated. In the case of chronic anxiety, I believe attribution is to be avoided rather than encouraged. Oftentimes you are experiencing dread without any real external triggers whatsoever, providing no readily available, preferably small threat that you can safely attribute it to. In such cases the mind tends to go toward generalized reasons such as job security, health, financial security, self-worth and the like, which really only serve to aggravate the situation, as these are not simple events that are temporally limited or that can be dismissed as insignificant. Transforming an already false attribution of “I am anxious that I will lose my job” into “I am threatened that I will lose my job” serves no purpose, or so it seems, unless it also helps in the realization that there are in fact no actual threats to your continued employment at that precise moment (the value of which would be dubious in any case, as your mind will then wander to find other causes for your feeling of anxiety). Maybe your solution works in cases where an over-exaggerated response to stimuli is the issue, but in cases where anxiety exists without obvious external causes, where false attribution is a bigger problem, I would still think that addressing the false attribution itself by realizing that your mind state has no cause is a safer approach.

This isn't obviously pertinent to the topics of this specific post, but the idea of chronic or frequent, and persistent, anxiety reminds me a lot of the ideas behind Internal Family Systems (IFS) therapy, namely that the anxiety is a strategy by which a 'protector', a cognitive and emotional part of you, is protecting one or more 'exiles', other parts that in a sense 'encapsulate' trauma.

The IFS practices seem, on their face, very different from CBT or (Buddhist) meditation traditions, but I suspect they're leveraging much of the same internal 'machinery' of the mind.

In a comment on this post you mention that you "replay the last few minutes, and usually I feel triggered again when I get to the original trigger.". That reads very much like IFS ideas about communicating with one's parts, at first protectors, e.g. a part that uses an "anxiety trance" to avoid exposing other traumatized parts to something negative, and then, with the 'explicit agreement' of the relevant protectors, the 'underlying' exiles.

I've had some success overcoming everyday fears by trying to be courageous in my thoughts and actions. For me it works better than trying to think my way out of feeling afraid. But I never had chronic anxiety and don't know if that approach would work there - have people tried it?

I'd offer a different question. And I'd suggest a reframe of anxiety. Anxiety is about the body delivering more energy to itself, it comes with extra mindful attention, and it's about protection yes, but not necessarily threat.

Most of the time when I get some sensation like anxiety I'm thinking about how I might benefit from this extra energy that my s1 has decided I need. How I might use it to pay extra attention and me more vigilant or cautious for errors.

As you said it's not really a threat, for me it's more about my concern that I'll make a mistake.

"anxious" energy is here to help me to be more vigilant and cautious about this concern.

Given the way emotions work there's a danger of false attributions as rpapp suggests. If I for example show a person with arachnophobia an image of a spider for 20 milliseconds they will feel anxiety but they won't know why they feel it. The image is enough to trigger the emotional stimulus but it's not enough to raise to cognitive awareness. The mental processes that produce anxiety are strong enough to match patterns that don't rise to cognitive attention.

If that person then goes and says "I feel threatened" and tries to reason about why they feel threatened, there's a good chance that they come up with another reason for why the feeling exists.

Debugging a false reason can still release the negative emotion and help you deal with it, but you shouldn't take it too seriously. Hold it lightly. It's similar to how past-life regression can help people deal with emotional issues. It's a technique that works, but if you take all the information that comes out of it as being literal truth you run into problems.

The correct mental stance is a light one of curiosity and exploration and not one of "I now have to accept the serious reality of how flawed I am".

In addition I would expect that you will be better able to deal with the feeling if you first feel into the felt sense as taught in Gendlin's Focusing.

There's are two emotional ways to react when threatened. One is anger and the other is anxiety. It frequently happens that people suffer from anxiety because they don't allow themselves to be angry because they want to be a "nice guy" or "nice girl". David D. Burns sees this need to be nice as one of the pillars of a lot of cases of people have anxiety disorder in When Panic Attacks.