An Apple TV will be competing with a lot of different things, and from my perspective it will have to be excellent at all of them to register more than a blip.
As a streaming box, the device has iTunes and Airplay support, so that's good. But compared to basically every device on the market, the current Apple TV lacks in apps and its interface isn't very good. For an Apple product? Damn. And compared to something like a Chromecast and Android, the Apple TV and iOS aren't sweethearts as much as they should be. So the new device needs to step it up, and stand out from the Amazon/Chromecast/Roku devices that everybody has.
As a gaming box? The device has to beat out a used Xbox 360. Looking online, that's going for $80 right now on Gamestop's website with 10 free games. Why wouldn't I buy that? If I were only interested in playing on my iPad I guess. I don't see the Apple TV changing that.
Apple sites ignore facts occasionally (I read them, I'm well aware of this) and one of the facts they ignore is that console gaming isn't doing that poorly. Mobile gaming's nice and big, but console gaming isn't on its death bed. The PS4's been selling like crazy for $400 - and it's because of the library, something I don't think Apple developers can replicate.
EDIT:
Thinking about this some more, the obvious comparison is handheld gaming killed by mobile gaming. While that's generally true (or at least handhelds were affected), I think that speaks more to the lack of features handhelds had and how much the mobiles brought to the table.
For an Apple TV, a PS4 will do just about everything better, aside from play iTunes content. And unlike the Nintendo console, I don't see it being a secondary console purchase for most gamers. I don't think there will be many Apple TV exclusive games. Maybe easy access to games like DOTA 2 and Warframe could make the Apple TV an interesting competitor for the single-game-focused-yet-still-frugal hardcore, but I don't think the current mobile gaming developer scene will deliver Apple a slam dunk like the Apple blogs think.