My theme provides a wpml-config.xml but nothing appears in the Strings translation table

This issue could occur with child themes when the parent theme stores its options in theme mods.

In that case, edit the wpml-config.xml file provided by your parent theme, and change the line:
<key name='theme_mods_child_theme_name_example'>

Then put this new wpml-config.xml file into your child theme’s directory, or in /wp-content/polylang/. You must create the directory if it does not exist.