Alright, I had been suspecting that the Fuel Solenoid as the problem, so I removed the bowl with the solenoid and replaced it with the old bowl with a straight screw-on plug. I used the NEW main fuel tube jet because it had a newer o-ring on it, and the bowl-to-carb gasket is brand new as well.
While it wouldn't start unless I primed the carb, the mower started, ran fine for more than a minute (a record for me, since it usually dies after 30 seconds), so now I'm certain the original problem was due to the Fuel Solenoid. I continued to let it sit running for about 4 or 5 minutes, while I cleaned up the mess of tools. I then decided to take advantage of the opportunity since it was mostly sunny out to actually get some mowing done.
Got about halfway across the yard when the engine started to sputter again, so I took the burden off the engine by coming to a stop, putting the gear in Neutral, and disengaging the blades. Immediately the engine perked right back up for about 10 seconds, after which it sputtered yet again and died. This was after continous good running for 7 minutes, the last two under load (in gear, and blades engaged). Now I'm really confused.
It cranks ok, but the engine still won't start-up when turning the key still, so I suspect there is still a clogged port in the original old bowl.
So I prime the carb and it starts and runs fine for a minute and thirty-seconds. This happens repeatedly.
Since the problem is now effectively worse, I took my old bowl off the new Carb again, then pulled the Fuel Solenoid off of my new Bowl and replaced it with the plain threaded plug from the old carb and re-installed the new Bowl back onto the new Carb. Still using the new Gasket and Fuel tube. Effectively, the entire NEW Carb is now installed back in, except that the Fuel Solenoid has been removed and replaced with a plain threaded plug.
It now starts right-up again without needing priming (so that would tend to indicate that my previous thought about the old bowl still having a plugged port is likely correct), but still dies after 1 minute to 2 minutes. So now I'm considering that when I replaced the bowl these last two times, maybe I mucked something up.
The first question I have is: the main fuel tube has a series of 3 micro holes (2 grouped together near the o-ring, and a 3rd hole about the middle of the tube) on two sides directly across from each other. Do those have to be aligned in a certain direction when placed in the carb bowl? I had thought that I had re-aligned them exactly how I found them on the NEW Bowl when I pulled it out originally, but maybe I got that alignment wrong.
Thoughts?