Translatable strings not appearing in the translation interface
As suggested by @prkos in the comments, visiting the pages (both public and admin) in the target language caused the corresponding strings to be discovered by Drupal and therefore become available in the interface translation. For each string, you have to visit the corresponding page (or block).

If you have custom error or confirmation messages, you apparently have to trigger them before they appear too. (? not entirely sure)

(I thought the "updating translations" progress bar that you see during the module installation was precisely Drupal listing all translations available in the module, but apparently that's not the case.)