Made any progress?
Am going to binge watch a few C64 repair videos tonight, I love murder mysteries. You have probably seen this before but checkout the link below. It mentions some books at the end, "Troubleshooting And Repairing Your Commodore 64" by Art Margolis looks fairly comprehensive.
http://retro64.altervista.org/blog/commodore-64-repair-a-quick-guide-on-the-steps-required-to-fix-it/The second link shows some other tools born of that time period the logic pulser and probe, clip and current tracer (there was also a logic comparator as well). In the logic clip video (he is demoing on a arcade board) you will see him disable the crystal and manually drive the clock with the pulser.
https://www.youtube.com/playlist?list=PL7K-x1E-x5j_XhmMCJg99_BhLusGDCUgTAside from the speed of testing on those tools, triggering on an oscilloscope, a function generator and or a logic analyzer replaces those (functionality wise) and some programmers cover the testing of logic. But doing something similar combined with the board running on minimal ICs (if possible) and a dead test cart then comparing to another board seems to be about as low level as you can get for complex issues.
Here is the troubleshooting approach I would take working within your specific rules of minimal board changes:
1. Proper confirmation that power has no issues and the whole board is supplied and grounded.
2. Video has been ruled out (as much as possible).
3. All input is non responsive and known points of failure like electrolytics, temperamental regulators, dry joints, degraded traces, corroded pins or bad sockets have all been addressed.
4. Basic signs of life and signal quality as per the last video I linked or web guides, service manual, datasheets.
5. Low level or minimal config troubleshooting compared to a good board with the dead test cart.
6. Manually testing and observing data lines.
7. In circuit analog signature analysis with a curve tracer in the hope it shows something missed.
8. Then lastly removing and swap testing components to check if you have an obscure IC or board fault.