Update: I finally have this HP 3456A now working.
1. I re-started troubleshooting from the beginning. To start, I replaced the old Nichicon capacitors on the power supply board. That cleaned up the rails quite a bit and reduced the ripple on the +33V rail by half.
2. I picked up an HP 5005A signature multimeter on ebay at a reasonable price, so now I can follow the service manual steps. I couldn't figure out how to interpret the scope waveforms for the microprocessor so there was no way for me to determine if what I was seeing was good or bad. But the signature analyzer is exactly what I needed. The service manual describes how to connect the signature analyzer to components and test points on the board and you touch a component with the test probe to get a digital signature. You then compare that measured signature to the signature value published in the manual; if the value matches, it's good, else it's bad.
3. Using the signature analyzer, the CPUs tested good, address bus good, clock good, etc... But line D2 on the data bus was reading a bad value. Following the suggested troubleshooting flowcharts, it suggested bad ROMs on the A4 board. However, the ROMs all showed a bad signature on pin 11 with all the other pins reading good. Having all three ROMs become defective on the same pin? That seems unlikely, and it just happens that pin 11 on all three ROMs correspond to line D2 on the data bus. So, some component connected to the 8-bit data bus was bad somewhere in the system.
4. The 8-bit data bus connects from the A4 board to the A3 board, the board where I saw the bad corrosion when I first opened the lid. So, I replaced chips U1, U6, U13, U15, and U17, since they were involved with processing the data bus and had gates used across the entire board in different places. No change. So, finally, I replaced chip A3U10, the SN74LS165N the 8-bit shift register, and powered it up.
Success! The meter came alive, passed the self-test, and started showing measurements!
The meter seems to have maintained its calibration well and it gives readings close enough to my Fluke 8505A and HP 3457A with my 5V voltage reference.
I guess I could have fixed it without the signature meter, but I would have probably replaced many more chips that were good before I found the bad one(s). But the Signature Analyzer does give a definitive answer of whether the CPU, memory, bus lines, and timing lines are good. It clears the innocent parts but it alone does not find the true problem(s). But it's a great tool to help you know where to look for the problem and not waste time looking where you should not be looking. At least, now I have learned what good waveforms look like in the microprocessor lines.
Also, assuming that A3U10 was the only problem, I could have found the problem more quickly had I strictly followed the service manual steps. Service Group A does say to replace A3U1 and U10 if the transmitter waveform pattern doesn't have the correct number and combination of highs and lows, 2 high, 3 low, 4 high. I was seeing 2 high, 2 low, 4 high, in the defective A3 transmitter output.
Finally, in this specific case, I could have solved the problem the moment I opened the lid, had I merely replaced all the chips on board A3 that had any green corrosion on their leads. Chips A3U10, U13, U15, and U17 all had green corrosion on their leads and traces. In fact, removing U10 and U13 exposed some significant decay of traces underneath the chips that were not visible until I removed those chips from the board. Please see the photos.
It seems that moisture had gotten on to the chips on the A3 board since A3 is exposed to the environment through the bottom grilled panel. Board A30 has the metal cover panel so that board was protected. The top side panel doesn't have vent holes on the cover panel so A4 and A40 were not exposed to moisture either. I shall be alert to attack corroded chips first in the future.
Thanks to all who provided me the helpful advice to get this boat anchor sailing the measurement seas again.