The latex editor example is the one that best demonstrates the concept. It is a standard JS frontend that is invoking an unmodified pdflatex for rendering. In all other 'OS in the Browser' projects I'm aware of it would have to be a desktop application that just happens to be getting rendered in the browser. With plain emscripten or webassembly you would need to modify pdflatex to support a method to accept jobs and return results while not relying on any OS level features.
Writing an operating system that directly targets the browser, that offers the expected/standard system calls ought to be smaller and faster, and more interestingly, it can potentially integrate with the rest of the page in interesting fashions, easily.