The complication of traction, the loading does make a difference, especially for
smaller objects (smooth surfaces tend to attract each other, which is probably
the main reason that the smaller block seems to have much more grip).

Look at the latter part of clip # 2:

http://www.gyroscopes.org/1974lecture.asp

The relationship between slip angle and side force is not linear with the load
factor, look at the graph of force versus slip angle and note that the curves
differ depending on the load (this is slip angle, not slip ratio):

http://www.smithees-racetech.com.au/ackerman.html