Update. Right before my last post, after it had sat for a couple hours, I started it again and it ran for 30 minutes. Sputtered or "missed" a few times, but other than that, it ran strong for 30 minutes. I decided to move to the next step and put it under load by engaging the blades. It ran for about 3 minutes with the blades engaged then died. STarted right back up, engaged the blades, ran for a minute, then died again. So I started it right back up and it ran for 10 minutes and is still running right at this moment.
So....minor improvement. At least the silly thing will run for a while. Guess I could drive it around the yard and mash the grass down.

Still it seems to me that it is getting starved for gas. Is there anything to rebuild or replace carb-wise other than just cleaning it?
Also - I looked over the fuel lines. No cracks, no leaks, all looks good.