Loading config overrides (e.g. translations) in listings should be configurable

I suppose, the option to list the config entities should be per user

Not necessarily.
By default, the config listing should use the language which makes more sense for the site. The current behaviour "without override" or the new "with override" option. If the current language does not work for some users, then those users should have the option to change the current language with the language switcher.

Not sure about your use case, but if there is a need for more granular setting, then I would prefer the option to be "per list". Like most of the config lists can be displayed with override, but the date formats are displayed without override (as that was the original issue when the config listing behaviour was changed).
Maybe there could be a switch to on every config listing to show the list with or without overrides.

I came to this issue while I was trying to fix https://www.drupal.org/project/drupal/issues/3030949 🐛 Config entity label and other fields translation doesn't work on some pages Needs review

It seems that current implementation done in https://www.drupal.org/project/drupal/issues/2136559 → has valid reasion and done on purpose.
Still if someone needs the changes, I've tried small workaround by using a flag to see if we can control whether we want to load an entity override free or not.
Let's see how test suite goes.

smustgrave → credited belyj_vladislav → .

Closed 🐛 Config entity label and other fields translation doesn't work on some pages Needs review moved over credit from original author who worked on it in D8

Not sure if the rerolls need to be brought over.

This can also be useful in non-translation contexts. For example, the domain → overview now lists all domains without config overrides. I however override domains in local environments in settings.php, and it would be useful if the current, local domain were shown instead of the default.