I don't get people who dismiss the idea based on the look of this specific demo. Use your imagination and focus on the core idea.
I don't get people who forbid Apple from implementing something like this because they don't want it and think it is a terrible feature. Most people probably have something about iOS they don't like even if they generally prefer iOS. Some people hate folders, so they don't use them. Diff'rent strokes.
I do not stare at the home screen on my iPhone, nor do I have any interest in doing so. That said, I would love to have a 2x2 block on the home screen showing my next calendar appointment or two. Not to stare at, but to have more opportunities to be incidentally reminded of something I'm supposed to be doing or need to do soon. This is the same reason Outlook has the ability to show your next few appointments while in the email view.
I don't know how many times I've unlocked my phone in a hurry to do or check something, inadvertently swiping away a meeting reminder that I didn't notice because I was distracted or away from my desk momentarily or didn't feel my phone vibrate in my pocket. If the next time I went to my home screen, a Calendar widget was there to show me I was 5 minutes late to a meeting, that would be fantastic. It doesn't require a separate intentional motion of swiping down the Notification Center.
Similar argument for a Reminders widget. I don't see much use for other widgets off the top of my head (I personally don't want the Weather widget consuming minute amounts of battery and data every time it is visible on the screen), but I don't fault others who may want other widgets. I think new uses would present themselves over time.
I think this concept would help ergonomics with an upcoming larger screen. Even with the current 4" screen, the way I usually hold my phone for one-handed access to the whole screen requires a slightly uncomfortable motion, or changing my grip slightly, to reach the Home button. If a widget occasionally saves me from diving into an app to look at one thing and then pressing the Home button to exit, it's worth it to me.
For the temporary widget idea, maybe a simple double-tap on the app icon could open a hovering widget view if the app supports it, then it goes away with you tap elsewhere, including tapping to launch a different app. Limit activation of permanent-widget status to the rearrange-the-wiggling-icons mode after pressing and holding an icon.