Hi Steve,

WordPress only checks on ‘upload_files’ to give upload permission. But that capability is also needed to give access to media library. As you said, you will be able to remove the “Add New” using menu editor, but the “Add New” within the Media Library uses ajax upload so it won’t be limited. So its not possible through the UI at this moment.

I could give you a code snippet that you can add to your functions.php to limit the user from uploading files if you are comfortable with it.

Thanks

Syam