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:

1. Update the content.
2. Build
3. Deploy

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:

• you are locked-in to a third-party vendor - you do not have full controll over your blog application. Let's say your provider fails, like this year's Amazone outage. You are powerless and can only hope for the best.
• performance can be a problem - serverless applications are known to have greater latency in how the compute resources respond.
• debugging serverless applications is tricky - although there are some tools, debugging serverless applications is a hard thing to do.