If you have a large weight above the center of gravity, and the stage gets off-vertical a little, that large weight now exerts a torque that pulls the rocket off-vertical some more. A weight at the top is only viable if the deviations you are trying to connect are very small (like a 100-m building swaying 1-2 meters in high winds). But the stage needs to be able to fly at large angles, as your video shows.
A large weight at the bottom of the stage would be better, and it's the only place where you already have a large weight, i.e. the engines. But to make that work, you'd need to make the engines movable relative to the rest of the stage. Given the huge forces involved, that would mean a heavy structure. The rocket only has a few tons of payload, this would reduced that significantly.
Your idea was looked at by SpaceX for the Dragon at some point: it was to use a movable weight to orient the capsule during reentry.