@1FAST4 Thanks for the informative videos.
So, it could be fuel starvation, but when I have experienced fuel starvation, the engine tends to start, run, then fade, and die, not fade and get better, and then fade and die.
The three things that I have seen do fade, better, fade and die are a) debris intermittently clogging fuel line somewhere (engine vibration can knock debris around), b) an air leak into the fuel system causing injection issues (and since yours runs ok for awhile, the leak is likely small), and c) a gummed up governor rack that isn't moving freely and sticks. Given the smoke on dying, I would tend to suspect option b or c.
I think that that style pump typically has two check valves per pump.
This storm does look to be a gulley washer for a lot folks in NorCal/Oregon.
All the best,
2Pbfeet