WebTV died years ago. I don't think many people really want a modern website on their TV.
I wouldn't mind twitter/RSS updates in a side area or Growl box (like a chat box in a FPS), but the web doesn't fare well on the lower-resolution TV; similarly as with mobile devices, websites would have to be redesigned for TV usage.
People may not want modern websites, but articles and/or documents can still be read not in the context of a standard browser. Twitter updates along a side is a good idea, perhaps email notifications such as comcast does with caller id too.
I wouldn't mind twitter/RSS updates in a side area or Growl box (like a chat box in a FPS), but the web doesn't fare well on the lower-resolution TV; similarly as with mobile devices, websites would have to be redesigned for TV usage.