in post-1 You do not write about the Replay-Counter and You do not write about the Coin-Relay.
I believe when You (successfully) start a new One-Player-Game the Replay-Counter does step down one step and the Coin-Relay pulls-in for a while. I guess when You try to start for a second Player: No stepping down on the Replay-Counter - no pulling-in of the Coin-Relay. Am I guessing right ?
See the JPG - starting a new One-Player-Game: The pin is not busy resetting (switch-red-1 is closed), the pin is not busy letting run the Score-Motor for some reason (switch-red-2 is closed), You press the Replay-Button (and close switch-red-3), there are replays on the counter (switch-red-4 is closed), the game You just have finished (the game before) has ended normal means You has reached Game-Over so switch-red-5/B is closed as the G.O.-Relay has tripped, current can flow to the coil (red-6) - the Replay-Relay pulls-in, establishes Self-Hold-Current by closing red-7, also closes red-8 and the Coin-Relay (red-9) pulls-in, actuates switches , closes (red-10) its Self-Hold-Switch. The Replay-Relay and the Coin-Relay stay pulling for a while - then (see arrow, red-11) a switch on the turning Score-Motor will cut the Self-Hold-Circuitry of these two relays - they quit pulling.
JPG, encircled blue: Pressing the Replay-Button has no effect when the pin is busy resetting, whenever the motor is running - also a thrown-in Coin will be reflected into the return compartement (Coin-Lockout-Coil non-pulling means reflecting).
A,B,C,D,E: These switches also hinder current to flow to the Coil on Replay-Relay - when You have started for FOUR players: switch-red-A has opened so You cannot start for an (not existing) fifth player. C, D (, E) are open when the pin has completed the starting of a new One-Player-Game.
The reason why C, D (,E) are in the pin: In a One-Player-Game we can press the Replay-Button when we are playing Ball-2,3,4,5 - we say: I do not want to play the remaing balls - I want to start for a new Game --- In a 2/3/4-Player-Game we can press the Replay-Button when we are in Ball-1-2nd-Player (or 3rd, 4th) - Ball-2,3,4,5 - we say: We do not want to play the remaing balls - we want to start for a new Game. In the JPG I added "marked brown" - some pins do have (Jubilee has not): A switch on Ball-Index-Relay - as soon as the very first ball has made the first points: We can press the Replay-Button and start a new One-Player-Game.
Your description in post-13 is good - ONE thing You did not test: The switch on the Coin-Relay - either the relay is pulling but the contact-points on the switchblades are oxidated - or the blades do not touch --- or the relay is not pulling. Greetings Rolf