Is it possible to use the Apple TV as a gaming console - and if so, how, and where are games available for it listed?
An argument can be made that they already have a gaming console: the iOS platform. Some games (The Incident comes to mind) already support streaming the game through an AppleTV, while using the iOS device as a controller. With iOS 5, there is supposedly built-in support for streaming almost anything through an Apple TV. My biggest critique of iOS gaming on my iPhone is that my fingers get in the way, so we'll hopefully see more streaming-the-screen-through-a-TV.
Now in 2020 it's totally possible to use your Apple TV as a gaming console.
Apple introduced Arcade which is a subscription service, that provides access to many games available on macOS, iOS/ipadOs and Apple TV (you can even transfer game progress between devices). It has much less games that you can find in Steam or in PS Store, but it's totally fine for casual family friendly gaming.
You can even use controllers with Apple TV now - Connect a wireless game controller to your Apple device.