All the cycles should function such that after each of the wash and rinse agitation periods: momentary pause of the motor, restart for neutral drain of two minutes (the motor runs but no agitation or spin), another brief pause, motor restarts for spin.
Fill
Wash Agitation
Neutral Drain
Spin (may or may not be sprays depending on the timer programming)
Fill
Rinse Agitation
Neutral Drain
Spin (probably always two sprays)
[2nd Rinse is a repeat of fill, agitate, neutral drain, spin]
The Permanent Press cycle has a cool down after the wash: momentary pause, partial drain (until the water level pressure contacts reset) with another longer pause for the remainder of the 2-min drain increment until the timer advances, refill (always select a cold rinse for Perm Press or the cool down has no purpose) and agitate two minutes, then proceed with a full drain, spin, rinse/agitate, drain, spin as outlined above.
Operation of the motor is electrical. The motor reverses direction for agitation vs. spin. The timer of course pauses the motor before a reversal so it first coasts to a full stop. The pump always runs with the motor -- spin direction it drains, agitate direction it pushes the water into the tub outlet instead of pushing it out the drain hose.
Neutral drain is a mechanical function inside the transmission, via springs and weighted cams that hold or allow the spin gear to rotate. A brief period of agitation is required to set the mechanism so the spin gear is held stationary at the next motor reversal. Another pause releases the hold, then restart in the same direction spins the basket. Setting the timer directly to a drain or spin position after a full cycle has ended will result in immediate spin since no agitation has yet occurred to set the neutral drain mechanism.
Seems your no-spin problem is electrical since the motor is not running (per the video).
Could be the speed selector switch is bad? Neutral drain is always at high speed so the motor circuit likely is hard-wired through the timer, doesn't go through the speed switch during drain. Spin speed is selectable so the motor current goes through the switch during spin periods. Agitation is a different path through the switch to allow different combinations of agitation and spin speed.
You didn't say if it spins when a low spin speed is selected.