I'd like to (attempt) to diagnose the problem if possible. Is this something I might be able to do with an o-scope, DMM and logic probe? I'd like to become more fluent in reading schematics and applying them with the tools I have. I don't want to call it quits yet because I've made it this far haha
In this case, those tools aren't really going to be sufficient for this level of problem. You need the ability to capture long traces of logic data, and compare them to a known working board. Or you'd need a logic comparator, which would let you compare the streams of data from one chip against another good chip in real-time.
You sometimes can narrow down problems like this with a logic probe, but it takes experience. And the ability to listen to and compare what you can hear with the probe, against another known-working board, side by side.
As I mentioned above, the easiest trick to try would be piggybacking the chips in the Vector Timer section, one at a time. I'd start with the four 161's. This is a useful trick for a lot of VG issues. But like any technique, it isn't a silver bullet, and isn't guaranteed to catch every possible problem. Part of working on these boards is having as many tools and techniques as possible in your arsenal, and trying multiple until one of them hits on something. No one tool works for every possible case.
Another trick in these cases is to suspect any chips that are Signetics brand, if they are in the section you are focusing on. These are labeled with a big 'S', or 'SA', or 'SB' on them. They tend to fail at much higher rates than other brands on these boards. Not that I advocate shotgunning, but if you just replace all Signetics chips off the bat, you'd stand about a 75% chance of fixing most VG issues.