Well having it actually not magically fix itself last night caused me to fix the actual problem today.
Here is the IGNITION SWITCH wiring diagram for my 80FI:

I have an aftermarket Lada Ignition Switch:

Given that there is a "click" heard, and that this problem happened both with my old original starter and my new gear reduction starter, my conclusion is that there is just enough juice getting to the solenoid to close it or click it, but not to engage it.
The RED wire from the "50" position on the ignition switch goes through C31, then through C6 and connects to the Starter. It triggers the solenoid to pull in the dreaded brown wire power at C50 on the starter. (C31 is actually a 5-position connector, it is a curious and somewhat irritating aspect of these purchased wiring diagrams that they don't show the complete portions of the subject connectors, but that is a discussion for another day).
C31 is a located a few inches behind the ignition switch, C6 is a 1-position bullet connector, located in the engine compartment down behind the alternator (next to the brown wire bullet connector). I verified that I had connection from the ignition switch through to the C6 connector, so some head scratching leads me to conclude that there is a connection, but it is weak and won't pull the solenoid in strongly enough to engage the starter.
C31:

Sure enough, it looks like C31 is a NON-LOCKING connector. It appeared to be very loose when I first inspected it. Don't ask me why they would do this.... Chianti again. Cleaned up the ignition switch connections, pushed it all back together tightly, same for C6, and no more "click of death". For now, anyway..... Maybe a tight ty-wrap around C31 would make some sense?
Just thought I would share this, as it has come up before in the discussions here.
Pete