Would you say that the game plays correctly, but only as a 4 player game? If so have a look at the Coin Unit (described at https://pinside.com/pinball/forum/topic/bally-old-chicago-won-t-score-properly-for-2-4-players#post-5490952).

The Coin Unit is probably a step unit with four positions. It should reset at the start of a new game and take a step forward as extra players are added. If your Coin Unit is stuck in the last (4 player) position, or if the path for a 1 player game (shown in the other topic) doesn't complete the circuit you'd only be able to play 4 player games.

Also it looks like the wiper fingers in your photos don't land in the center of the contact pads below them. That could cause problems if the wipers contact two pads at the same time.

/Mark

What happens when player 4 ball 1 drains? Player 1 ball 2?

Either way, I'd suggest that you sort out why you can't start a game with the replay button first. You can't be sure you're getting a complete reset otherwise.

The reset sequence is described in the game manual on IPDB.org. You should be able to start a game by stepping up the Replay/Credit Unit and pressing the Credit button. If not have a look at the Credit relay circuit (on the schematic also on IPDB).

Unless there are some signs of rework that I didn't see it seems pretty unlikely that there's a wiring issue on the player unit.

Here is a picture of the player unit from my #old chicago.
IMG_20200829_225917 (resized).jpg
I don't think the wiring would be incorrect because it doesn't look like it has been reworked.

Here are some other ideas.
1. Is the ball index relay energizing after the first score for a ball? The behavior you describe doesn't sound like this is the issue, but I have to ask.
2. Is there a bad contact in the Jones plug from the mounting board to the upper cabinet? I spent a day chasing down a similar problem to find a corroded connection in the Jones plug want making a good contact.