The Bounce switch (cab board), Anti-Cheat switch (coin door), and Shutoff switch (cab board) all need to be normally closed. If any of those are open, then the R relay can't energize and hold.
But from a powered-off state, R won't energize on it's own. What needs to happen first is the SB relay needs to trip, and a make/break switch on there completes the path to the transformer from the line cord. Once this path completes, the 25V side of the transformer is able to activate the R relay, and a switch on R itself closes and keeps the power held on (thus being the 'Hold' relay).
So if you can manually trip the SB relay and have R energize, then the thing to debug is getting the SB relay to energize and trip.
To get SB to energize, it needs the same make/break on itself, a normally closed switch on W, a normally closed switch on the ball gate switch stack ("On Ball Release"), and then either a path via the replay button or the coin mech switch.
You can see this logic path at the bottom of your schematic.
As for the power cord, here's a reference for the 2-prong style:
But if you're able to manually activate R, then you're getting power as you noted.