EEEK!
Something was miswired in the cabinet.
You have some 74LS273 chips and a 74LS05 along with some resistors making up a resistor ladder on the video outputs. Check the resistors and replace any that are open or have drastically changed values. Replace the 74LS05, fix the burnt traces, and then see if you have video output. If you don't, then change the 74LS273 chips.
I can do this for you if you aren't comfortable working on a board. I charge $35 for most non-surface mount Neo-Geo repairs.
And thanks for the pr0pz Cheffo!