Rudder
I agree with you, the rudder will take a long time to properly analyze.
I flew the 737 and remember all the training we got about rudder hardovers. NONE of it included an on the runway encounter. I did imagine what I might do, and it is making the centerline your friend. AND DOING whatever it takes to stay on the centerline.
I recall the tragedy of USAIR 427 and the United COS crash. I can imagine a scenario where an unusual side load on the rudder, either meteorological wind (cross wind), wake turbulence, or something similiar triggering a rudder event.
HOWEVER, as I mentioned before, staying on the runway is a vital part of the takeoff. AND I know our airline has always admonished pilots against using the tiller above about 20 knots or so.
AS I recall, a reduction in hydraulic pressure happens at 700 feet radio altitude...but no protection on the ground. The first and last minute of flight is always the most challenging. I always imagined what I might do below 700 feet to keep from hitting the terminal or other planes. As I said in my first post, assymetric thrust can be your friend.
I am not ruling out other causes including human error, but nothing is off the table just yet.