Yesss, finally a use-case for the `<noframes>`-hack! This is actually even better than the approach of OP since we're guaranteed to get a plain text representation of the text (instead of the browser parsing it as HTML).

example.html:

    <script src="mdpage.js"></script><noframes>

    # Header
    Welcome to my simplest site

    - A
    - awesome
    - list

mdpage.js:

    document.write("<div id=main></div>");

    document.addEventListener("DOMContentLoaded", function() {
      var script = document.createElement('script');
      script.src = "https://unpkg.com/showdown";
      document.head.appendChild(script);

      var noframes = document.querySelector('noframes');

      script.onload = function() {
        var converter = new showdown.Converter({
          emoji: true,
          underline: true,
        })
        converter.setFlavor('github')
        main.innerHTML = converter.makeHtml(noframes.textContent)
      };
    });