If you broke a ring I wouldn't be surprised if it gouged the cylinder too. then you would still have poor compression and that'd cause the problems you're having
lotta detail in that page including plugging the decomp. bad decomp could cause the mentioned issues too.
Timing i'd agree though is most likely. you should be able to adjust through the windows in the flywheel.
do you have a 20mm Mik on a stock 50cc? I used to have an 18 on my sachs (I didn't set it up that way) you'd definitely be better off with a smaller carb on a stock jug. or go ham porting it up