Server side Drupal install as git local repo
Fair warning, I don't know Drupal, so it's possible you're describing a common workflow. All the same I know a thing or two about git, and what you describe is a bit backwards. The problem is here:


  
  The hosted Drupal install is my git's "local" repo.
  Make changes, commit
  Push to github
  


By treating your production environment, the hosted Drupal install, as your local repository, you basically circumvent the benefits of versioned deployment. If you break that local, you break the site. If you want to work on a test feature branch (maybe new content you're still creating), you have to expose it to the world. A more canonical approach might be:


Commit to the site locally (on your computer) as well as a clone on your hosted site. Use github to push and pull between them. The Drupal docs have a solid example of this.
Commit to the site locally (on your computer), and use a post-receive hook to deploy the Drupal site, if your hosting provider supports git installation. This is a common setup in web application deployment.


In either case, note that you want a truly local copy, some place to experiment around, rather than the live website.