PCB Repair: Out Zone
Problem
Game does not run; displays garbage.
Diagnosis
Another Out Zone to look at, this time one I purchased for myself as non-working:
In cases such as this, I first check to see if the CPU is running; in this case, it’s a 68000. There was a 10MHz clock signal present at the clock input and both /RESET and /HALT were de-asserted/high. There was no activity on the address bus however and /DTACK was low, which seemed a little suspicious. I would expect /DTACK to be high if the CPU was not currently performing a memory access.
There are schematics for Out Zone, courtesy of jammarcade.net. There are several sources of /DTACK generation:

Probing the relevant input and output of the 74LS04 at 1A revealed that the inverter was in fact, not inverting:

In the end, it took me longer to desolder 1A than it did to find the fault. The power pins were predictably tricky to desolder and I had to resort to using the dual iron method (soldering iron on the component side pad and vacuum de-soldering iron on the solder side pad) to prevent damage to the PCB. With a new 74LS04, the game ran as Toaplan intended.
Fix
Replace 74LS04 at 1A.