Earlier quoted context omitted.
Have you tried GitX? What you described seems a bit more like GitX than what GitHub is trying to do here. When I first fired this up, I made a tarball of one of the repos I'm currently working in (it has uncommitted changes) and started clicking around. I'd do something in the GitHub app, then go back to my bash prompt to see what happened. I was pretty confused when I switched branches and all my changes were gone.…
What's the canonical gitx repository these days? Everyone I know uses a different fork, much to my dismay.
Notably, SourceTree (what mainly use) and Tower (if the incomprehensible one-repo-at-a-time limit isn't a dealbreaker) now compare favorably to gitx in most ways.
So I don't recommend gitx anymore, but from what I hear Gitx (L) is a fairly active and popular fork: