Here's a video of Sonic 4 Episode 2 that demonstrates this: http://www.youtube.com/watch?v=8fUP6mh3aOg
Plus, touchscreen games don't all require you to tap specific points on the screen (as in your Reversi example). Look at Temple Run: you duck and jump by swiping down or up, from anywhere on the screen, rather than tapping on-screen Duck and Jump buttons.