Bystarter can be removed and tested. Disconnect the 2 wires that power the bystarter. Remove the 2 screws holding the bystarter retainer clip to the carb . Pull out the bystarter (it may help to twist as you lift upward; there's an oring in there, do not damage it). Connect the 2 leads to a 12v battery. the bystarter will begin to heat up and after a minute or two the needle/barrel will begin to extend, probably about 1/8". If it extends, the bystarter works. If the needle does not move, bystarter doesn't work.
Another problem could be clogged passage to the bystarter fuel well or clogged idle jet or any number of things. But if the scooter always starts when you starve it for air, then the problem could very well be inside that carburetor. Did this scooter sit for any length of time?
Also, which screw(s) is stripped?