Litt;
The video helps a lot but no answers yet. This one is going to be challenging.
It looks like your game has gone into a playing loop that is missing a signal to stop between balls and player numbers. I don't think it is a start up problem alone.
Is this video with the ball in the outhole or not? What happens if you play the ball?
1) Both cars do move at times, and stop at times
2) It looks like the game is on the 2nd player a lot but the scoring is mostly from the 1st player
3) The ball count does go up, erratically, (and the back light going off and on indicates this too)
4) Does not go into game over after 5 balls
5) The car motions seem disconnected from the ball in play actions. They don't seem to act together, although lap scoring is done correctly.
In general the game acts confused!
The problem could be almost anything, usually a bad wire connection, dirty or mis-adjusted switch, bad stepper or contacts on stepper, among many other things unknown.
For a novice the first items to do are the easy ones:
Unplug the machine from the wall!
1) Gently tug on every wire connection in the game to see if any come off or are already off. Tweezers or long nose pliers can be used.
2) Clean every brass rivet on every stepper (alcohol and a rag)
3) Manually activate the step up and step down (or reset) levers on every stepper to see if the stepper is sluggish or stuck
4) Pull off all Jones Plug connections (including the small ones inside the front of the game for the coin door and launch button). Also check that all Jones Plugs are actually plugged in.
5) Check Jones plugs for loose connections or bad socket connections. Re-install by pushing on and pulling off a few times before finally seating the plugs into the sockets. Double check the connections for continuity with a multimeter or continuity checker.
6) Check the outhole switches under the playfield. See if they act correctly when the ball is both in the outhole and out of the outhole.
Plug the game in again, put the ball into the outhole, and see what happens.
Hopefully something different happens, maybe good, but not always.
If it still doesn't work then it gets more challenging for a novice. This is where things can go really wrong if done incorrectly.
At this time someone needs to check all relay coils and stepper solenoids to see if they are good.
All switches need to be cleaned (abraded), and adjusted for proper gap.
Consider reflowing the solder on questionable solder joints.
Look at Jones Plug connections (both male and female) real close to see if they are actually contacting.
Check all wiring runs for worn away insulation creating short circuits.
Check switches for bent tabs that short out to another tab.
In the end what usually happens is that you find one single thing that was bad and you get aggravated at how many things had to be checked before stumbling across it!
Alan