I have one of these.  A 1961 which was the first year of the perforated tub.  Prior to that they functioned as a perforated solid tub, meaning that you had 3 seperate tubs -- a perforated liner bolted into a solid tub that spun the water over the top and into an outer tub.  The agitator column does not "move" even though it appears to in videos, on the earlier models it has more "movement".  The real movement is in the bottom flapper.  Inside the agitator column is a lint filter/softner/detergent dispenser.  On the later models (when Philco was owned by Ford)  the column has no movement as it appears in the last video you posted.  From what I understand these models do not have as good a turnover as the earlier ones.  The agitator moves at all times, even in the spin cycle.  There are a number of posts here on the site that explain the mechanical operator in great detail.  Somewhere there is a photo thread that Robert posted of a rebuild on one of his.

I like my Philco because of it's unique and rare nature, not many of them are around -- we hear less about them than we do Kelvinator/ABC.  It washes, rinses well, and spins adequately (550 rpm)  but I would not pick it over one of my Frigidaires.

  Inside the cabinet I at first thought this was oil spray, turns out that Philco sprayed rust protection inside the cabinet walls.