You have a lot of issues, and narrowed them down. One major issue seems to be the carb going lean indeed, due to a malfunctioning high speed cirquit? That explains the early advance running better with small spark plug gap (which retards combustion onset).
Beg-steal-borrow a good carb, or bring the carb to your nearest garden appliance shop for a checkup. (if they have good craftsmen there!). Be sure to tell them your findings and history.
You also can order a new carb, and save you a lot of hassle, budget permitting. Your problem seems to need a major carb overhaul that most are not willing to do (included me, I just do not have all the required tools).