Yeah but that would seem to defeat the purpose of having a screen of that length for display, if it's essentially always going to be cut off that drastically.
There's just no getting by it. If it's actually a 6" screen (which must again be stressed is goddamn huge to be on a controller), and they want to display the streaming stuff (and whatever other media) at a workable aspect ratio, the screen will have to have some decent height to it held horizontally, not just length. And that's going to mean a fairly large controller sitting in your hands.
None of the rumors seem to have mentioned some sort of flipcase or slider mechanic so I assume the screen sits on top of the controller and is visible at all times, which would also make sense if they're trying to put game info on it or use the touchscreen to "enhance" gameplay.