Amazing that they were able to figure this out. I have a PET that also sometimes boots up into a garbled screen similar to the one pictured in the article. I am usually able to get past it with a few hard resets. I don’t have access to (or much knowledge of) logic analysers, but this almost makes it look doable to figure out the problem.
If you want to go that route, you may find it easier to check your logic analyzer dump side-by-side with the debugger in an emulator (e.g. VICE). Break on the first instruction and then go step-by-step. (Just need to make sure you have the same ROM version, but that shouldn't be too hard.)
However, your problem kind of sounds like a power supply problem. So using a logic analyzer will maybe just produce a different result every time. So maybe check the 12V and 5V rails on an oscilloscope while turning on the computer. (Or maybe it's a problem with the reset circuit, etc.)
Thanks I will see if I can get my hands on an oscilloscope and look at the 12v and 5v rails.