Once again an article posted to get clicks... This "rumor" is so blatantly false its sad, but it sure gets the readers in a frenzy. Even Kotaku states in their own article, that they have had multiple Durango owners tell them that an internet connection is not required to play games. You never see that linked though.
The nintendo Wii was "Always on" for the same reason the next xbox will be. To download updates while the system not being used instead of forcing the user to download updates when they just want to play a game.
As for the disconnect statement. I dare anyone to play world of warcraft, diablo 3, battlefield multiplayer, simcity and disconnect their internet and see what the game does. Under the same light, try starting a game without internet connectivity. OMG the game didn't work, lets start a huge rumor about how the xbox sucks...
http://en.wikipedia.org/wiki/Sheeple