Whoa, never seen that screw before. I want one of those carbs to make in to a 14! Is it a 12 or 10?
If it tries to start only when You add premix directly then it sounds like You have a fuel delivery issue, go through the carb again, make sure to clean out the idle 'jet' passage.
If You're getting spark, fuel, and compression, it should start....
You can check the timing with a timing light (You might need somebody to pedal or use a drill), that would save you pulling the engine and flywheel to inspect everything, or maybe even check on the key with a flashlight if you take the clutch and spacer off, never tried shining a light down the crank, but there's a gap so maybe. It should not come loose off the taper and slip/shear the key without a puller or at least a mallet though. I doubt any shock strong enough could be transmitted through the belt, maybe if something jammed in the flywheel fan blades while the engine is running. (edited)