Does that url redirect to a different url?
Can you check the browsers console to see if you get any javascript or network issues?

Can you give me (super) admin and (s)ftp access so I can take a look?

You can use the 'Confidential information' button in the forum editor to hide sensitive information.

No, you can communicate via this forum.

Try using the url it redirects to instead.
So: /en/?option=com_...

Seems to be working as far as I can see...

I opened that page and saw "Testing 123". I clicked on the Cache Cleaner button, refreshed and saw "Testing 456"...
🤷🏻‍♂️

If you can't get it to work, then it seems that that xtdir4alg component refuses to clean its cache when it is called through php. Maybe some security thing in there. Not something I would be able to fix for you. I recommend you ask them why it would not do its thing when called via Joomla\CMS\Http\HttpFactory::getHttp()->get() method.

The way it is set up now, it should clean it when saving too (when Cache Cleaner is triggered).
If that doesn't happen, I would need to dig into it and would need (s)ftp access to be able to do tests in the file system.

Well, the url is correctly being queried when saving an article. The url gives a 200 (OK) response, with:So Cache Cleaner seems to be doing what it should.
If calling that url (via Cache Cleaner) doesn't result in what you expect, you will have to contact the developers of that component.