OK, I fairly certain the two wires that have solder on the ends belong to the total play meter, which apparently is missing. The jones plug socket to the left that does not have a jones plug in it i believe is the ball count adjustment. There should be a wire hanging somewhere with a male plug on it that plugs in there, from left to right it increases to 5 I believe, I will have to check when I get home.
Did you solder or desolder anything at any point? Solder splash could be completing the circuit across any of the switch stacks that are associated with the 10 point reel that you have listed, i would visually inspect all of them. Also, this is probably not it, but those rollovers, if the gap is not adjusted properly, when the playfield is down, the switch could make. Its unlikely but worth noting.
Lastly, i noticed in this picture:
https://img-f.pinside.com/201411/2031157/316109.jpg
The stepper unit on the right, seems to be misaligned. The fingers of that outer plate need to make contact with only one rivet at a time. There also looks like there could be bridged rivets on that unit, cant tell for sure by the pic, but if thats a solder bridge across the rivets, that could be your issue right there.
I am in the process of restoring my 4 Roses at this very moment, if you need pics of something in particular, I could send them.

This is what it should look like and where it comes from.

image.jpg

What is different between yours and mine is the jones plug that is not plugged in on your game, if you trace back to the unit above as shown in my pic, you may find the plug or wires. Thats why I posted the pic.
Did you check the rivets and disk alignment on that right stepper unit I suggested earlier? That right one is the yellow advance unit, and could very well be your issue.

Sorry, i didn't see that video. In this game, the rollover buttons advance the steppers as the yellow and red rollovers are made respectfully, so if the stepper is stuck between two sets of rivets, or if two rivets are shorted together, it could be the source of your problem. That stepper (the yellow one, on the right) could be gummed up or may have reached the end of its cycle at that point. I am not in front of the game now as I am at work and i don't remember, but if it is a continuous stepper, that one is for sure gummed up and needs disassembled cleaned and lubed (which should be done on all steppers IMO anyhow). If it is not continuous, it has reached the end of its "step up" or "step down" and the other coil will step it down when it resets. Either way, it looks to be misaligned and looks to have something across the rivets in that one pic that could be shorting them together. Read this:

http://www.pinrepair.com/em/index2.htm#steps

Section 2d will walk you through it, don't skip any steps, the steppers can be the difference between the game working and not working. Be sure to mark the disk with a sharpie in the "reset" position so you put it back together correctly.

Correct. I will check for sure when I get home in a few hours, and probably shouldn't say until then, but i'm fairly certain i'm on track here. For sure manually pulling in or pushing out the plunger on the coil on top of the unit, directly below the label that says "yellow advance unit" should step the stepper in the opposite direction. I don't think this is a continuous stepper, they usually do not have clock springs, and for this particular circuit and application it wouldn't make sense for it to be continuous. It's purpose is to advance the yellow (and the unit on the left the red) rollover to the next position as it is achieved. Advancing the lights for the gobble hole on the playfield, the rollover lights, and the score advance. So according to my theory each stepper unit should step up/down 5 steps. I will check for sure when I get home.

Okay, they are not continuous steppers. They step about 20 steps each. Mine are pretty gumed up too, so cant tell how many exactly until i clean them. I have not gone through this game mechanically yet. The top solenoid, when it pulls in, resets the stepper in one pull. There is a pin on the back of the cog that stops when it hits a plate, so that will tell you when it is completely reset. You can manually pull that solenoid in and turn the disc toward you until the pin stops it if it is gummed up good, which both of mine are. Just about every game I've worked on needed the stepper units serviced in some way. Especially the Williams games. Just follow clays guid and you will be fine. Don't forget to count the amount of winds on the clock spring so you know how many to wind it back when reassembling.
Lastly, there is also a switch that when the stepper unit reaches the end of its rotation, that switch closes. If you're unit keeps trying to step like it does in that video after you clean it, that switch may need to be adjusted as well.

If you have a rotisserie, sure why not. I just do all of my work with the pf in the service position though. Its really not that hard, wont take more than 20 minutes each one, just read that section on clays site thoroughly so you have an understanding of how they work before tearing into it.
I am assuming you mean the two relays directly next to the steppers, if so, they are the red and yellow advance relays, the red is on the left and the yellow on the right, same order as your stepper units. They should pull in and advance the stepper via one of the switches in that stack, so yes they would fire at the same time as their respective stepper. Above those relays is the pop bumper on relay I believe.
Also, check the coil and all associated wires on the stepper unit on the left (red), if it is not even attempting to fire or step, there may be an issue with that. Those steppers being attached to the bottom of the playfield are dangerously close to the transformer and other units on the bottom board when pulling playfield forward and out.

You really need to disassemble those steppers and clean them as stated above quite a few times. There is not much point in trying to figure out what is stopping those coils from firing until you free up what it is they have to move. After you get them operating freely, I can look into it further as my game is almost done.

If it would make you more comfortable, i could take mine apart and take pics the whole time, as mine need to be done anyhow. No promise on time frame though, Im working like a dog the next few weeks.