The Whirlpool transmission repair video above is only the neutral drain kit, and spin clutch which is external to the transmission. All the other parts in the transmission were not discussed and/or removed.
This video covers more of it but still not quite everything.
6:00 the neutral drain plate is correctly called the rack retainer. The screw anchors it to the main drive gear so the rack retainer rotates with the main drive gear whenever the motor runs. The rack retainer is also present in spin-drain versions of the transmission, on which there is a spin pawl but not the other two neutral drain components, and the underside of the spin gear has a spring that fits on the hub instead of the toothed cam.
6:12 the spin gear pinion is driven by the larger plastic spin gear. The notched top of it protrudes through the transmission cover (an oil seal is there) and the clutch drum mounts to it. The spin gear is held stationary during agitation and neutral drain so the spin gear pinion and clutch drum accordingly also do not rotate.
6:22 is the connecting rack and is what makes the agitator shaft oscillate.
6:44 is the shift actuator and is what shifts the agitator gear cam to engage or disengage agitation according to which direction the motor is running.
6:51 is the main drive gear, which meshes to the input worm pinion/gear beneath it.
7:00 notice that the agitator gear cam parts are plastic.
7:06 the disc and ball comprise a thrust bearing for the agitator shaft.
7:25 the input worm pinion/gear (to which the motor coupler mounts) is not removable unless the oil seal on front exterior is removed.
9:49 the first neutral drain piece is the trip lever which is involved in resetting the mechanism during agitation for neutral drain that then follows.
10:11 the second piece placed on the rack retainer is the spin pawl which is what drives the spin gear.
10:16 the third longer metal piece is the latch which holds the spin gear cam from rotating during neutral drain, which in turn holds the spin pawl from driving the spin gear.
The spin gear cam is shown at 6:03 in Eugene's video above, on the underside of the spin gear. The flat end of the spin pawl mates to any of the three protrusions (bosses) on the underside perimeter of the spin gear to drive it. One of the three bosses (the one at the bottom by his thumb) has a little button molded into it which bumps the trip lever and is involved in the neutral drain reset process during agitation (10 bumps does the reset).