Doesn't sound like the carb is plugged, HMMMMMM...........When pulling over with the cord I think you are rolling the engine over with more RPM than with the drill motor, Which tells me that the no start/flood condition is caused by insufficent spark as a direct result of low rollover RPM. Remedy: use the pull start - or- tune it up, which would include new spark plug, inspection of coil pickup and magnets on the flywheel. If the magnets are even a little bit rusty, clean them up with a wire brush. Re-gap the coil to a smaller tolerance to the magnets.( If possible)
If you are dead-set on removing the pull start and the tune-up does not help, There are electronic ignition kits available. (Unknown if they have one for your engine) If you can find one, they could be spendy. Hope this helps, Good luck.