So if you put the reels to say 1111 and press the start button, it resets the reels and the score motor stops running?

A clue in the video shows the bonus ball count unit advancing. A hole relay or a score 100 and advance bonus relay is probably stuck closed by either a closed hole switch or the relay hold circuit switch is closed on the relay.

One technique that is sometimes helpful is to watch what relays de-energize when you turn the machine off; take a slo-mo video if needed to spot them.

Ok find the advancing relay on the schematic and look what switches activate it and check them out.

Just throwing in the tired and seemingly useless advice:

Have you attempted to do this with a ball in the trough? I’m not familiar with these 60s machines, however, these games are usually designed to activate only when a ball is in the trough ready to be served.

It certainly seems like there is an issue with the bonus count being stuck somehow. But you can never underestimate the power of “make sure there’s a ball in it”

It’s a phrase just as powerful as “have you checked the fuses with a meter”! Live by it!

Check the advance relay for normally open switches that are closed when they shouldn’t be. Also inspect switch wire connections for shorts.

Normally 5 balls.

No. Only one for add-a-ball games!

So you're basically saying that there's no power to the playfield?

It's difficult to see on the schematic you provided. I'd look at the game relay, tilt relay, and game over relay. Engage/disengage them manually. Clean the contact pads with an ignition file or light sand paper. Check that each switch on those relays changes state when you manually work the relays. You may have a misadjusted switch that cuts power to the playfield. When a switch closes, the longer blade will push the shorter blade that it contacts slightly, showing that it is making good contact.

I'll need to see a high quality scan of the schematic from (for example) Staples ($2), Fedex ($6), or Office Depot

You could look at the color of the wires going to the coil, then match with the wires shown on the schematic.

The schematic in post#1 is legible when expanded. After examining the schematic and your video I’ve determined that the trough rollover that is being pushed is trying to pulse the tilt relay latch coil. When this coil pulls in the playfield power is restored, and the score motor is running because of a stuck closed playfield switch or the reset relay didn’t switch over power to the playfield. Once you release the rollover the score motor stops because the tilt relay is not latching, or it’s trip coil is pulling in and not let it relatch. Also the tilt latch coil fires during the reset cycle. Inspect why the tilt relay isn’t latching.
The way the trough switches are set up my guess that this is a 2 ball in trough machine.

Releasing of both balls to start is normal. After that alternate balls will be captured and released in the trough.
Can you post a pic of the mechanical board to help identify tilt relay.

The tilt relay is a trip/latch type so one on the bottom board (probably) with two relays -- an M-29-1000 (latch) and M-29-1000 (trip).

The tilt relay latch coil should have a yellow-black and a black wires connected to it.

The tilt relay is one of the 2 coil relays in between the transformer and the score motor.

Is the TILT light on in the backglass? Trying to see if it is being latched, as that is required to get power to some of the coils.

Same with game over.

You want to check the manual operation of latching hold and trip coil releasing of the relay.
Watch the relay during reset of the machine to check if it latches also.

And make sure you have a known-good bulb in the socket.

The relay you're manually activating is probably the Tilt relay. Maybe you can tell by wire color, or maybe you can get another Wing Ding owner to post pictures of the original relay labels. In any case, ignore the highlights I made for the Game Over relay, and here's the circuit that latches the Tilt relay:

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Slow motion video from your phone can be very helpful. Sounds like the tilt relay is latching but then immediately tripping -- a video would be helpful to confirm that.

On the tilt relay find the switch with the yellow-red wire and a jumper going to the trip coil.
Insert a piece of paper in between this switch to keep it from closing and retest the reset cycle.
Remove paper and see if the trip coil pulls in immediately.

In your 2nd video, when the relay closest to the transformer pulls in its latch coil, it doesn’t latch because the plastic part of the latch is worn down allowing trip armature plate to slide away and not latch without the trip coil pulling in.
The other latching relay isn’t adjusted correctly.
The trip armature plate should be pushing downward more on the latch plate, changing the state of the switches, and releasing when tripped.
Adjust trip coil assembly for proper operation.

Good basic overview of Relays, Coils, and Switches

To adjust latching relays:
Loosen 2 trip coil housing screws to where the housing can slightly move.
Manually push in on the latching armature plastic ladder until about it reaches about 80 % of its maximum travel.
Move the trip housing so that the trip armature is in its latched position with the other armature.
Tighten down trip housing screws.
Release pressure from latching armature.
Latching armature should stay in latched position with little movement.
Manually inspect latch trip operations
Check or change switch gaps for proper latch and trip positions.

In the 1st video, latch position is A.