Well, finally, after splurging on a genuine Stihl/Walbro carb and finding that made no difference whatsoever, I gave up on repairing it myself.
I took the saw in to my Stihl dealer on Monday.
They called back this morning, to say it was fixed! Ran great!
And all they did was adjust the magneto gap.
From a technical standpoint of fuel+spark+compression=runs, I guess it was the spark. But not only had the saw run more or less fine for those 1-3 seconds each time I'd try it, but I'd checked the spark with my spark tester (multiple times). I gather that the spark tester is useless at distinguishing between a weak spark and a good one, but I had no idea a small gas engine could consistently run-and-die due to a weak spark. I have always believed that if the spark is too weak to stay running, then it's too weak to start.
Guess not. Live and learn.
I had replaced the magneto a year ago, due to an outright broken ignition wire, and I did at the time have trouble setting the gap. But in the end I'd believed I'd gotten it right, and the saw ran fine then. And I can't imagine that vibration since then would enlarge the gap; that magnet has a lot of force.
So it was not a crankcase leak, or low impulse-line pressure, or a maladjusted carb, or bad rings, or cylinder scoring, or any of those weird things.
I picked it up this afternoon. $35.