What happens when you hit the credit button on the game to start a new game?

Your running score motor when the ball drains problem could be explained by the Advance Unit not resetting. Here are the relevant bits of the schematic annotated for each of the steps described below:
Skylab Advance Unit (resized).jpg
I believe that the sequence should be:
1 - The ball drains and closes the Outhole switch.
2 - When the Outhole switch closes it fires the Bonus relay. Note that neither the Advance Unit nor the Ball Count unit are at zero so their switches (to the left of the Outhole switch) are closed.
3 - When the Bonus relay fires, a switch on the relay closes to send power to the Score Motor.
4 - The Score Motor starts turning because the switch on the Bonus relay is closed.
5 - When the Bonus relay fires, its holder switch closes and keeps the Bonus relay active, and the Score Motor running.

I think this is what you're seeing and why the Score Motor keeps turning. What you may be missing are the remaining steps:

6 - The Bonus relay stays active through its holder switch until a switch on the Advance Unit opens when the Advance Unit steps down to its zero position.
7 - When the Bonus relay is active, it closes a switch that allows the Advance Unit reset solenoid to fire and step the Advance Unit down to its zero position.
8 - A Make/Break switch on the Score Motor (which is turning) closes once each time the score motor turns 180 degrees to send a pulse to the Advance Unit reset solenoid through the Bonus relay switch (#7).
9 - The relevant Score Motor switch is a Make/Break switch at the top of the switch stack on the #6 score motor cam.

The Advance Unit is probably a large stepper under the playfield or on the bottom of the cabinet. I think it steps up one step at a time as bonuses are accumulated, and steps down one step at a time when the ball drains as bonuses are added to your score. Your Advance Unit seems to not be stepping down. This could be because it's mechanically bound up or because the circuitry driving it isn't working properly. Here are a few things to try:
- When the score motor is turning, manually push in the plunger to the Advance Unit reset solenoid and let it go to see if the stepper steps down. If it does step down, repeat until it reaches the zero position and see if the score motor stops turning.
- Carefully check that the Bonus relay switch (#7) and the Score Motor switch (#8 & #9) are working correctly.

And to answer a question in your video, hitting the left flipper button to light up the game is normal.

/Mark