I figured it out. The smell comes from the phenolic wafers that separate the contacts on all of the relays and the stepper contact board.
I have that same smell in my 1957 Bally ball bowler, and an even stronger aroma that's almost identical in my 1953 United puck bowler. It's the same as my 1948 United WISCONSIN pin.
I picked up a relay rack off of Ebay as a spare, put it in my office, and after the weekend in there, the whole office smelled like the other machines. That rack has 16 relays, and a ton of those phenolic spacers, so it's GOT to be the source.
I wish they made a candle with that scent...I'd keep it burning in the game room all the time.