Hello Søren,
I'm happy to chime in. I tried publishing a sample Storyline course to the Web, and my course works as expected in a web browser. I see the same file structure as you.
Can you please try double-clicking the story.html file in File Explorer and seeing if it opens a browser so you can view your course as expected? If the course loads normally, then this is how it should appear once you upload the published Web output to a Web server.
If the course does not load normally, would you be willing to share your Storyline file with us so we can look at what you have set up? You may also share it privately by sending it along in a support case. We'll delete it when we're done testing