1) Pull the fuel line off the carb and do a visual flow test (1 quart in 1 minute would be an average benchmark for a 1/4" ID fuel line and a full tank)
2) Try a new spark plug (Yes, another one)
3) Other than that, it's the carb - as you stated, half choke or adding fuel make sit last a little longer
I doubt you are losing spark, you COULD be losing compression (unlikely but good to check valve clearance) You could have a plugged muffler (Think Banana in the Tailpipe A La Beverly Hills Cop) - engine MAY have a spark arrestor, which if that gets clogged with carbon, will cause the same symptoms.
Now that I think on it, plugged muffler would explain ALL of your symptoms and tests so far. (It would quit even with adding fuel) - OR, your fuel line is bad - Keep in mind they are a multi layer, the inner layer COULD be collapsed causing a restricted fuel flow - or could be routed in such a way that it pinches (but then, adding fuel *should* typically overcome that)
It is unlikely a coil problem, or a valve hanging up, as you say it starts right up after it quits (Which again can point at plugged exhaust) - a coil breaking down from heat would not do that. Likewise a failing coil, unlikely to allow it to start right back up immediately same with fuel line restriction - again everything seems to point at a plugged up exhaust.
Even if there is no spark arrestor, an internal baffle could be loose or broken and clogging up the works..
So I would go with - Exhaust, Spark Plug, Spark Plug resistor cap (plug boot), carburetor, in that order. - Mainly because checking exhaust, a new plug, and a new plug boot are the easiest, fastest, and cheapest to check first.