That's the Control Motor. (It's usually called a Score Motor in a pinball machine, but for some reason it's called a Control Motor here.) There's a schematic on ipdb.org that will help sort this out. Here's part of the schematic for the Control Motor:
Action Baseball controll motor (resized).jpg
The Control Motor turns when any of the switches in the red boxes close. To figure out why your Control Motor keeps turning you'll need to figure out which of these switches is closed. Keep in mind that they could close because the relay or motor that controls them is firing, or because they're stuck closed, or there's a short, etc.

First check if any of the relays that control those switches is firing while the Control Motor is turning: Start relay, Out relay, 3rd Out relay, 25 cent relay and 10 cent relay. If the labels on your relays are missing you'll have to use the wire colors on the schematic to find the relays in your game.

If none of those relays are firing, check the switches to see if any are stuck closed. You could stick a piece of folded paper between the contacts of all 6 switches to see if that stops the motor. Then pull the paper out one by one to see which one starts the motor.

The CM '0' switch in the red box is the top switch on the switch stack on the Index cam on the Control Motor closest to the motor itself. That's the switch that keeps the Control Motor turning until it reaches the home position where it should open and let the motor stop unless one of the other switches is closed too in which case the motor keeps going.

/Mark

That's enough to keep the Control Motor running. Next, figure out why the Start relay isn't relaxing.
Action Baseball Start relay (resized).jpg
The Start relay switch on the lower path in red is probably closed since the Start relay is active. Is the Game Over relay switch stuck closed or is the Game Over relay tripped?

Please read reply #9 again carefully. If your Start relay is stuck on (and making your score motor run) you need to figure out why. The Make/Break switch on the Game Over relay (in red on the schematic in reply #9) could be stuck closed and keeping the Start relay on. The state of the Game Over doesn't really help until you can verify that the switch on it that keeps the Start relay on can open.

If you move the Game Over relay back and forth while the Start relay is on, can you get the Start relay to let go?

Yeah, coils rarely go bad unless they've been overheated. But we're making progress working back from the running Score Motor. The Score Motor runs because the Start relay is on. The Start relay stays on because the Game Over relay keeps it on. Now we need to sort out why the Game Over relay won't toggle when you start a new game.
Action Baseball Game Over Latch relay (resized).jpg
Since you can manually toggle the Game Over (interlocking) relay and get the Start relay to let go, let's assume that the switch is working and look at why the relay isn't behaving correctly. The switch in question is a Make/Break switch inside the red box above.

The schematic says that it is drawn with the interlock relays in the latch (not the trip) position, and in that position the switch to the Start relay is open. Since yours is closed I think that the Game Over relay is stuck in the tripped position. So we need to sort out why it won't go back to the latched position.

The circuit for the Game Over relay latch is shown above. For it to fire at the start of a new game six different switches all need to close. Starting from the left side those switches are:
- Inn Unit 0 position: closed only when the Inning Unit is at 0 innings, open otherwise
- 1st 1 PT DU 0 MK - a switch on the 1st player 1 point score reel that closes only when the score reel shows a zero
- 1st 10 PT DU 0 MK - a switch on the 1st player 10 point score reel that closes only when the score reel shows a zero
- 2nd 1 PT DU 0 MK - same as above for player 2
- 2nd 10 PT DU 0 MK - same as above for player 2
- CM5 - a switch on the bottom of the switch stack on Control Motor on cam #5 (sixth cam from the motor itself)

Somewhere between the Yellow supply line on the left and the solder lug on the Game Over relay latch with the black-white wire on the right is either a switch that won't close, or a loose connection or some other kind of open circuit. If you don't know how to narrow that down have a look at https://pinside.com/pinball/forum/topic/big-brave-bonus-problems/page/2#post-4484030 or ask for clarification.

I think that's the right stepper. The schematic shows one solenoid has red & red-brown wires, the other has black & blue-red wires.

The switch you're looking for on the Inning Unit connects a yellow wire to a yellow-green wire. Your 2nd photo shows what looks like a yellow-green wire in the middle of the left side. I'm going to guess that the wiper is tied to yellow and that the yellow-green wire is tied to the rivet in the 1st or reset position. If that's the case it could be that the wiper isn't connecting to the rivet. You mentioned that the spring was unwound. Does it reset all the way at the start of a game?

How did you verify them? Individually or all together? Unless you can measure less than an ohm or so (with the power off) between the yellow wire on the left of the schematic and the solder lug of the Game Over Latch relay with the black-white wire on the right of the schematic there's an open somewhere. Keep in mind that in addition to the switches, the solder joints, jones plugs, etc. all need to be good.

Alternatively, you could use a bulb tester to see if your relay coil is getting power.