Has this clutch always had this problem or did it slowly develop/worsen over time?
The engine idle speed is set to its lowest setting before the engine will die.
Does it sound/seem like the engine is idling slowly/normally?
The reason I ask this is you probably have no tachometer for reference and I've ran across engines with a defect that would cause them to die if I attempted to set the idle speed below 2000rpm. In this situation, a perfectly good clutch would give the exact symptoms your describing.
After every hour or so of operation, 3 to 5 drops of motor oil dripped between the drive gear and the retaining bolt/washer at the end of your crank will usualy be sufficient to lubricate the clutchs bushing/bearing. As your oiling, rock the kart back and forth a little so the drum turns. This will help the oil will work its way in.
(be aware that over-oiling can cause the excess oil to work its way into the shoe/drum area causing the clutch to slip)
Without re-lubrication every hour or so of operation the bushing/bearing could possibly be seizing to the shaft and causing your symptoms.
If your chain has ever been ran excessively tight, your clutchs bushing/bearing may have already been damaged. This scenario could possibly explain your curent symptoms too.