Why would you expect "mouse-free control" for an application whose user experience is fundamentally designed around a pointer?
Of course, GitHub for Mac should leverage Mac OS X's Accessibility infrastructure so it works with assistive technologies like VoiceOver and full keyboard control (I hope it does!) but I suspect from your comment that's not what you're looking for.
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Why would you expect "mouse-free control" for an application whose user experience is fundamentally designed around a pointer?
Of course, GitHub for Mac should leverage Mac OS X's Accessibility infrastructure so it works with assistive technologies like VoiceOver and full keyboard control (I hope it does!) but I suspect from your comment that's not what you're looking for.