Figured it out! I wasn't wiring the first 24VDC coil's N.O. contact to latch when triggered.
Now it works!
Pressing one momentary (N.O.) button latches all other 120VAC relays open and lights the lamp associated with that button's circuit until the momentary (N.C.) button across the DC(+) is pressed, releasing the latch on 24VDC.
Pretty smart thinking on his part. He is actually using SPTT relays for the 24VDC coils switching 120VAC and SPDT relays for the 120VAC coils switching 24VDC so he only needed 6 relays total.
Hey, E.C.! Can we get some more options for relays please?