You shared the copy of your site that is 5.5 GB and its too much to download and setup
I've created a test site where it seems I am able to reproduce the issue when I setup the site with WPML with WPML default language is non-English like your site.
I've reported the issue for further investigation to our next level support. Please hold on for further updates.
I just checked with my test site where I've added the button block directly to a page and then translated the button text using string translation and it works:
=> hidden link
[You can change the language from footer]
Then I've run another test where I've added a page and within that page I've added a block view and within that block view I've added button block and added then translated the button string from string translation. I see it used to work but as soon as I add new block, the button string is not get translated when I change the langauge. I am not sure yet if I've added the block exactly how you added it.
I will try to grab the duplicator copy from your site and will try to troubleshoot the issue further next week. It seems this needs deep debugging.
I will get in touch with you with my findings as soon as possible.
I got the news that we are moving to new translation workflow where we should use the Translation Manager to translate the string rather then going to String Translation to translate the strings.
Can you please setup a test page for me where I can run few tests OR is that OK if I create few test page on your site to run few tests?
I have set the next reply to private which means only you and I have access to it.