Can airliners land with auto pilot in gusty wind conditions?
To the first part of your question, the airplane is not moving BECAUSE his yoke movements are &quot;strong and fast&quot;. Experienced pilots can feel a gust and respond with a control input to counter it before it has enough time to affect the aircraft. (You really can't appreciate this from a camera fixed to the airframe.)  As mentioned in a comment, this is called &quot;staying ahead of the airplane&quot; or &quot;flying by the seat of your pants&quot;. Quick and correct reactions are required to keep the approach on track when being buffeted around.
Another thing that is worth pointing out is that the magnitude of control inputs is higher at slower approach speeds because of the reduced effectiveness of the control surfaces. You wouldn't want to move the controls that much at cruise speed. Just like when driving on a bumpy road you move the steering wheel a lot more than when cruising on the freeway.
I have never flown an airplane with auto-land capability, so I cannot answer the second part. However, given the experience I do have with a couple different autopilots, my belief is that even the best systems would be stressed trying to keep up while landing in very gusty conditions. There are just some things you still need a trained and experienced human for...