Also, we prefer to give priority to the translators' needs: are you really comfortable with making the wiki less easy to use for the translators/refugees, just to make it potentially easier for the volunteers you mention?

For the translators there is no big change: When you go to "translate this page", you still see the selection of available languages as autonyms. This change only affects the display of languages in the "other languages" box, not for other language selections.

This is for scenarios where the main user group are people who don't even speak the translated languages, but need to use the translations.
Typical example: A voluntary refugee organisation wants to translate some relevant messages (e.g. where to find a doctor, how to separate the waste etc.) into the languages of the people coming using the Translate extension. The translators are refugees, but the main users of that wiki are the volunteers not speaking these languages at all, but showing these translated messages to the people they're working with. Most of the volunteers are not able to read and distinguish the autonyms of e.g. Urdu, Arabic and Persian - for them it's much easier if the language names are displayed in their language.

I think this is the same issue as https://phabricator.wikimedia.org/T145295

Yes, I emptied the table translate_groupstats and that fixed it.
Now also the page translation progress indicators (I mean those "100% translated" tooltips in the "Other languages" section) are correct again.
This seems to be the workaround solution whenever inconsistencies made their way into translate_groupstats

I tried that in different ways but it didn't help, situation is still the same. Any other suggestions on how to fix that?
(PS Sorry for my slow response)

Actually I just realized that because of this bug the translation progress indicators also show wrong information (for most of the languages it says "0% translated" but actually they're completely translated)

Yes, I just checked with api.php?action=query&meta=siteinfo&siprop=statistics again: The job queue is definitely empty but the problem is there.