Purpose of the post is to save someone time who is having similar issue(s). So, I'm working on a 2009 Vespa LX50 4t. Scooter was running poorly at idle and stall out when attempting to give throttle. Scooter did sit for a while this winter. Took carb apart and ran it through ultrasonic cleaner and blew out all passages. Also ran all jets through and made sure they were clear. After assembling, scooter still ran poorly. Being how the carb was clean and I could see light through all the jets, started looking for air leaks. Found this at the manifold. Bike still ran bad after replacing the manifold, just not as bad. Looked at the spark. Plug was fine, but the spark was weak sauce. Tested the pick up and the resistance was off substantially. Installed replacement stator/pickup combo. Scooter ran better. Had a good idle, but now I can clearly hear and feel a lean condition when giving throttle. Checked float height in and out of the scooter, all good. Check flow from the tank, petcock good. Check to make sure passages and jets in the carb were clear, all clear. Checked to make sure accelerator pump was working and I was able to confirm it was. Checked compression and got 160psi three times, so engine is healthy. The scooter just kept crapping out when giving throttle.
What I eventually noticed was that if I could get through the dead spot, there was some top end. I look around my various carb bits in the garage and found a Keihin 35 slow jet from some CB400F carbs. The stock one in the Vespa is a 35S. Only noticeable difference is the total length between the jets. The part post thread was the same length, but the part above the threads was much shorter on the 35. Stuck it in and the scooter ran perfect. There was good pull through the RPM range and no dead spot.