Well, it's fixed. The original Apple /// that started this thread. I received a donor board yesterday that's in so-so condition, but good enough for chip swaps, so I replaced all four of the PROMs near the CPU as well as every IC on the board that looked better / less corroded than mine. Probably about 30 or 35 chips, in total.
Powered it on and it was in the same non-functional state as before. Hmph.
As a Hail Mary, I swapped the 6502 CPU and that fixed it.
So, the original CPU was not working properly and that caused the power on state to look like this:
I pulled and cleaned
each and every chip on the board, cleaned everything with alcohol and Deoxit several times, installed a replacement memory board and new all 5v RAM chips, eventually swapped corroded ICs and... no change until I finally swapped the CPU.
I'm glad it's fixed - currently running the Apple /// demonstration disk from its slightly squeaky internal floppy drive.
This is the only computer I've been so frustrated by that I actually started to dislike the machine. I wanted to toss it, but at the same time could not give up, hoping the solution would arrive eventually. I'm disappointed that it was something so simple, too. I wanted to feel a sense of accomplishment after all that, like "Whoa. Can't believe I figured that out." But now I'm thinking to myself "Oh... duh. Should have done that sooner."
I know some others have encountered this issue when powering on the machine. Some have fixed it with swapping PROMs. Try the 6502 CPU, too. Save yourself the trouble.
Fixed 256K unit running on the right.
To add insult to injury, just after that photo and as I finish writing this update,
the thing just popped another RIFA cap! I thought I changed all of them in its power supply...
I do not love the Apple ///.