> right click on an item, click "copy public URL", paste it into the browser, and get an exact copy of that item
You have described Google Drive.
You have described Google Drive.
Put <img src="foo.jpg"> into an html file, alongside foo.jpg. In the original Dropbox, if you opened a link to the html file, you'd see a webpage that successfully rendered foo.jpg. So you could use it as a static file host.