Why does Backdrop filter blur my entire app

It turns out the video is missing an important part from the documentation:

If there’s no clip, the filter will be applied to the full screen.

So the solution is to wrap the filter into a ClipRect, as taken from the example:

ClipRect(  // <-- clips to the 200x200 [Container] below
        child: BackdropFilter(
          filter: ui.ImageFilter.blur(
            sigmaX: 5.0,
            sigmaY: 5.0,
          ),
          child: Container(
            alignment: Alignment.center,
            width: 200.0,
            height: 200.0,
            child: Text('Hello World'),
          ),
        ),
      ),