What the hell is wrong with this thing

If you're running a petcock they're notorious for restricting fuel flow because the rubber inside them swells from ethanol exposure. I've also seen a handful of fuel pick ups fail. I dealt with a couple kawis that I had your exact symptoms and a pickup fixed it.
 
To the op: next time you have your e box open, check the resistance between ground and the brown w/ white tracer wire coming from the stator. Charge coil, part of stator can "age" and give you that 10 minute problem. If less than 510 ohm at room temp (new about 545 ) I'd replace it. Regrettably this static test is not 100% reliable, but ok.
 
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