Xbox 360 in the tail years (post 2010) and Xbox One from the get go, have good feature addressing TV, content and apps. Actually, XB1 is perhaps the only device that you can feed live TV from antenna or cable through it.
The original Xbox was sold to MS execs as a way to capture the living room. The whole idea of a set-top media streaming box was invented by the Xbox homebrew community, with the Xbox Media Center(now Kodi).
Xbox One was supposed to be the culmination of the vision. Too bad Microsoft alienated their audience (gamers), by mixing messages and designing their whole interface around a hardware fad, the Kinnect. Didn't help that they also alienated so many independent developers and which left them with a dearth of software as well.
The Channels application on Apple TV (http://getchannels.com) allows watching OTA signal from HDHomerun devices on AppleTV. Nice for a cord cutting solution so you don't have to switch from one input to another. Requires additional device (hdhomerun), but can be done. I've heard beta will allow things like pause, etc.. of live TV.
