I'm having same issue: starts and idles well, but when I apply throttle, it bogs down and stalls. By sticking my finger into the carburetor intake while at idle (until I can feel the fuel entering carburetor) and applying throttle, I am able to get it to run normally. This tells me the engine is running lean (not getting enough fuel) at higher throttle and thus stalling. By restricting the air being introduced into the combustion chamber, I am essentially enriching the fuel/air mixture.
If you can stick your fingers right in the carb, it doesn't sound like you have the airbox in place. The airbox typically poses a significant restriction in airflow, so the scooter may run normally with this installed.
If I'm mis-understanding you and you're just sticking your fingers in the airbox intake, then I agree your engine isn't getting enough fuel and an examination of the carb jets is likely in order.