Gottliebs starting in tilt is 100% normally on initial power on. It's just a funny quirk about gottlieb ems.

If you put credits on the machine and press the start button the machine should spring to life. If not there might be an issue with the path of power to the S relay, so start there.

Before ordering parts I'd watch a video on how to adjust the AX relay. If the plate seems sluggish, it can sometimes mean that the blades aren't putting enough pressure on the plate to move it back into position. I had the same issue on the bx relay of one of my games and a little switch adjusting fixed it.

I recommend trying what this guy says first. (relevant part starts at about the 3 minute mark)

From what I can see on the schematics, S should de-energize once when 2b opens, which happens on the last part of the motors 120 degree rotation, so that's working as intended. And yes it looks like the AX relay should keep the score motor turning. Keep fine tuning that AX relay, if you haven't already it might not be a bad idea to clean the switches as well. They can be adjusted properly, but if they're too dirty no amount of adjusting will fix it.