By default Media module comes with 4 default bundles Audio, Document, Image, Remote video, Video. And in code bundle "File" is mentioned in the media create function call. Also the default target field is also incorrect. It does not exist by default.
On fresh Drupal 10 setup with this module.
Setup the key and select save as Media checkbox in the augmentor page.
On testing (Preview) you can see it will throw an error of
Error: Call to a member function getSource() on null in Drupal\media\Entity\Media->getSource() (line 143 of /app/web/core/modules/media/src/Entity/Media.php).
Either mention in the Readme to create a Media type reflecting the code.
Or Map it to existing media type.
The patch i have provided uses Audio Media type to store the generated file in Media.
Review the patch.
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The File bundle is base type and is created by the Media module, that's why you can't see it listed in the media types page "/admin/structure/media", the error reported may come from a different issue.
Please share how you setup the augmentor field in your entity type.