I don't get your problem with automated testing. In my eyes this is the one feature that the shell has, but Github doesn't. That's why I'm quite confused. You can addd git hooks to automate it, or simply call the corresponding command while you are in the shell already. But GitHub neither builds nor tests your code, it simply shows you the git log, git diff, etc. as HTML.
Oh, sorry, I guess I wasn't very clear. I was talking about automated integration testing, i.e. testing on a wholly different system than the developer's machine. (This is to eliminate the "well, it works for me!" problem.)