Well, I zapped myself this afternoon.

:grumpy:
And I'm mad at me for being a doofus. I could have been seriously hurt. :nono:
The problem wasn't the solenoid. "The boy" (my stepson) and I installed the replacement yesterday morning. Still no joy when I turned the key. Nothing happened. Nothing at all.
We continued troubleshooting and discovered that the "Y" junction covered by the blue heat shrink (circled in red) was in bad shape, causing an intermittent open. The ground comes from the wiring harness and one lead of the "Y" is connected at the top of the solenoid mount (red arrow) while the other end is connected to the negative side of the solenoid. This is the original setup which had been working normally previously.
The "Y" connection was much worse than it looks in the pic.
One little tug on the "Y" connection and it came apart.
We cut and cleaned up the "Y" connection and reattached everything where it needed to be. Still no joy.
So, I did a test and jumped the negative side of the solenoid directly to common ground on the starter mount and turned the key. The engine cranked! Then, I put the shifter into gear and the engine didn't crank, which was good. The neutral interlock was working.
So, the starter, solenoid and neutral interlock are working.
We then made a trip to buy some connectors and when we got back, we rigged up a semi-permanent connection. The idea was to get the engine running and then dig into the problem this winter when the weather is cooler.
Got the temp harness made and connected. I left the ground from the harness open. (CIrcled in red.)
Then I pumped the fuel bulb, choked the engine and cranked. It started right up! I let it idle for a couple minutes, then turned the key off. But the engine kept running!
So I grabbed my alligator clips, attached one end to that open ground from the harness and started to put the other end on the common ground at the starter.
And that's when I got zapped! :facepalm: Duh, me.
I then pulled the fuel line and let the engine shut down that way. Then I reattached the fuel line.
Next, I pulled the kill switch and turned the key. The engine immediately fired up again!

Once more, I pulled the fuel line. Then we called it a day and went back in the house.
And that's where we are now. I have no idea WTH is going on. :noidea: