OK, real quick.
At some point before I posted here, I had already cracked open the carb and cleaned the pilot jet. I also verified the float needle was working correctly. And I've verified fuel is getting into the float bowl.
The choke is not a traditional choke, but it's an enrichment circuit. The fact that it runs when the choke is "on" indicates it probably is getting fuel fine.
It feels like it's like intentionally shutting off. My current theory (which I can't verify) is that it has a "ignore the sensors" start up mode that expires after a few seconds. I need to actually time how long it will run each time. If it's an actual time that's consistent, then it will bolster my theory. The theory is that it starts in this special startup mode, then after some fixed time, the PCM evaluates the sensors and then finding it's "not running", cuts off the ignition. The crankshaft position sensor is used to fire the ignition so it won't run or start at all without that. But the throttle position sensor could be bad and cause this? Maybe?
OK...
I just went out and tried it again. It started and ran with a very fast idle, probably 2500 rpm, for about 45-50 seconds, then idled down and stalled. It would restart immediately but just barely runs for like 2 seconds and stalls like the idle is set way too low. I cranked the idle up a little bit at a time and tried restarting it and it tried to catch a couple of times then refused to even catch at all and the battery went dead.
So my theory that it's an electronic cutoff is probably wrong. It might be that the electronic choke was never working right to begin with, remember I worked on it not long ago and though I had fixed it, and maybe it's working but just for 45 seconds and the idle is plain old too low, but I can't get it to run long enough to set the idle speed. You would think the idle speed would not have changed with it sitting all winter, and that at least I can adjust that and get it running if that's all that's wrong, since it's a thumb screw that I can easily access.
Anyway. I have a new carb on the way, and that certainly can't hurt.