Why complicate the whole process when you can use a simple static generator like Jekyll or Hexo (or any other static generator). My website and blog is using Jekyll and I'm more than happy with the whole update-deploy-build process. I write a blog post, add it to the repo and Github rebuilds the website. Almost any static generator works the same way:
Since you are serving a static HTML pages, you can use every possible hosting service you like.
While serverless applications have many advantages, they come with their drawbacks: