I usually just use a logic probe to start. And I'll piggyback chips that I suspect might be bad. I rarely use a scope.
Did we determine the board isn't watchdogging? Is it playing blind?
You can lift pin 11 of the LS139 at K5 (VGGO) to disable the VG. If you do that, and it plays blind, the issue is most likely in the VG.
You just have to probe around and figure out what sections are working and which aren't, then try to narrow it down. But as I said, it's one big loop, so you can't trace the logic linearly. Many times you'll see large sections of chips be inactive, if there's a bad chip anywhere in that loop. But you can start by looking for activity at the X and Y DAC's, and tracing backwards from there.
I'll usually start by probing there, as well as the Vector Timer, and State Machine sections (see the schematics), and get a big picture view of whether the whole VG is dead, or if any of it is working. Then take it from there. But debugging the VG is more of an art than a science, and it's one of those things you get better at with experience, the more of them you do. It's not something there's a simple flowchart for.