It pulls up the system keyboard - this widget seems to create it's own in the notification center.
The whole "no keyboards" thing has a very real, legitimate reason - keyloggers. As has already been mentioned, if a widget doesn't use the system keyboard (as whatever keyboard the user chooses is highly sandboxed and scrutinized before being allowed) the potential for security threats exist.
Apple doesn't want that. You don't want that. It's not that much more difficult to simply open a note taking app. It's actually extremely easy.
I mean at this point, why have a full app - let's just take all the functionality out of them, cram them into a small window/home screen and call it good. None of those horrible, painful extra taps to open the app and get a far better experience.
I concur with others here - if widgets are THAT important, go Android. Period.
You make a good point about a custom keyboard. And I agree with you on that respect. But if that's the security concern, then why can't Apple just let the widget pull up the system keyboard, just like the Quick Reply widget does? Wouldn't that solve the fear about the key loggers? Then it wouldn't be any different than the any other notes app inside the phone that opens the system keyboard and lets you enter a note, IMHO.