Comment by yoz

The File System Access API can do this, and it works in local ("file:///") HTML files, but it's only currently supported in desktop Chromium browsers: https://caniuse.com/native-filesystem-api

However, every browser will let you download a new version of the HTML file and save it over the old one - yes, even from a local HTML file:

   const blob = new Blob(['<!DOCTYPE html>\n' +
      document.documentElement.outerHTML], 
      { type: 'text/html' });
   const a = document.createElement('a');
   a.href = URL.createObjectURL(blob);
   a.download = 'mypage.html';
   a.click();

... but you need the user to do this for every update. So we're back to manual "Save" buttons.