The_Great_Man
I'm also an engineer and debugging problems on an EM is fun.
There seem to be a couple other wires going from the tab of the bulb socket that is giving you trouble to other bulb sockets. Are those working?
A couple more things to check in addition to the Z-relay.
1) Reflow the GND connection at the base of the bulb socket, where it gets screwed onto the playfield. When you manipulate the socket to get the new bulb into it, if you had a cold solder joint there, that connection could open.
2) Check to see if the bulb socket is good. Use alligator jumpers and bring the 6V from one of the playfield GI lights that is working to the tab on the bulb socket (where the yellow-blue wire attaches - don't connect it to anything else or you will short 6V to GND and blow the fuse). If the bulb lights up, then you know it's not the bulb socket and the problem is with getting power to the bulb socket. Which leads back to the "Z" relay that schudel5 mentions above.
Hope this helps,
Alberto