"Needing" depth really depends on your interpretation. Yes the github buttons don't need depth to be buttons, but at the same time, their depth probably makes them feel more clickable, and draws more attention.

Yes, I wonder whether you could use this as an extra emphasis effect for certain buttons on a page, have the important buttons (sign up, buy, etc.) with depth, and the less important - but still necessary - just flat. Would be interesting to A/B test.

Having thought about it, a fair few websites already do this with certain confirmation dialogues. eg. They have the primary button with a prominant graphical style and the cancellation option as just a small standard html link.