If the new AppleTV uses at least an A8, it can drive a 4K display without any problems. An A8X would be even better of course. Maybe Apple will use their A8X salvage parts and disable the third core (like they did with the A5), so it has a dual-core A8X in the new AppleTV. I would actually prefer this over an A8, because the A8X has a lot more graphics performance, higher memory bandwidth and should also be pretty power-efficient when you disable it's third core.
Some very sensible assumptions in your post. The A8X would have 4K 60fps video covered but I wonder how it would cope with rendering 3D at that resolution, as it would be expected to if the next Apple TV has an App Store.
Of course it could up-scale 1080p graphics but the option to render graphics at 4K would give console owners an incentive to get an Apple TV, if only for 4K 2D games (which could look amazing).