Thank you, Samuel! You helped me a lot to understand how it works.
You are only partly right though. It turns out that you can insert an image from the media library and the plugin still works as long as the image is not yet attached to another post.
In my case, I uploaded a bunch of images in bulk using Media>Add New, so they didn't get automatically attached to any post. That way I could insert them into new posts and your plugin worked. I just didn't realize that when I trashed a post, its featured image remained attached to the post in the trash, so the plugin couldn't work. I had to permanently delete the posts from the trash to make those images unattached and so the plugin work again.
Maybe it would be a good idea to update the FAQ accordingly.
Thanks for the great plugin!