Again, as Rolf points out, check the configuration of the player unit switches. Missing nylon stacks, but more so adjacent adjacent switch stacks.

Checking the switches for proper operation includes the adjustment of the switches. That will solve your problems.

I don't know your game exactly, but there may be another smaller stepper behind the chime box on the main board. I would look there first.

If the 2 and 3 player relays are both pulling in I would check for bent and shorted solder tabs/connections at the score motor and relays

The two seem to be acting in unison, so there must be a short. Can you jumper the contacts at the score motor and see if they both still actuate? Is there a separate contact stack at the score motor to close the 3rd player relay or does it use the same stack? If it uses the same stack, then the 3rd player relay should not close until the 2nd has.

My guess is that the closed 2p relay tells the 3p to close. What I meant by jumpering is simply closing the contacts at the score motor that affect the relay. Maybe you can use a small screwdriver. This will show if the 2p relay and 3p relay are isolated from each other.

Very good work. This is a main reason why I like to work on EMs. Most problems can be solved by your power of observation. That is, visually checking that the switches are adjusted properly. You learned a lot.