Chaining up to the parent's implementation of pick() is not enough: we
need to paint our children explicitly because of the compatibility mode
checks we use to avoid breaking custom containers.
https://bugzilla.gnome.org/show_bug.cgi?id=676088
We need to clip the children during picking as well as we do when
painting, to avoid reactive children outside of the visible area
receiving events.
This also allows us to refactor some common code into proper functions.
