Worrying clearly has a function, or else we would not have evolved to do so. Worrying motivates action - it is a countervailing force to laziness. Worrying draws conscious and subconscious attention to problems that need solving. It is necessary, like pain. How do you know you can't change something without thinking about it? Why would you think about it if you weren't worried about it?
I charitably interpret all the anti-worrying propaganda as an exhortation to try and maintain worry as a transient state that goes away when you commit to a course, as opposed to an ongoing coping mechanism.
[0] https://www.amazon.com/Peace-Every-Step-Mindfulness-Everyday...
1. Can I do something about it right now? If so, do it, and stop worrying.
2. Can I do something about it later? If so, wait til later, do it, and stop worrying.
3. There is nothing I can do. If so, there's nothing to worry about because I can't do anything about it.
Jesus said, "who of you by worrying can add one moment to their lifespan?" (or in another saying "...add one inch to their stature?")
The problem as you correctly put is that people worry, but that does not lead to a solution. There is really no point in worrying - if you cannot change something, why worry about it? If you can change it, why worry? You might have to think deeply about how to change it and act on it, but just worrying is like praying - it gives an semblance of action without actually doing anything.
The hamster, the wheel and the cage are illusions. We choose certain things as more important to us than others. You can always quit and walk out, but a lot of times we prioritize our comfort, an (illusory) stable earnings etc. over freedom.