I haven't worked on your bike but making an educated guess from having worked on other similar peds.
Look at the picture with the shaft that the starter clutch is on. Look at the shaft and the small thick metal collar with a slot or notch in it. This notch or slot is there for a reason, it is supposed to interlock with a "woodruff key" that is inbeded in the shaft. A "woodruff key" is a half moon shaped small piece of metal that fits in a small cutout in the shaft.
It is possible and sometime happens that this "woodruff key" can shear off or break, thus allowing the sarter clutch to spin freely with "catching" to start the engine.
Figure out how to lossen that belt and then slide the starter clutch assembly off the shaft to see if the woodruff key is slipped out of place or may be broken or missing.
You can buy woodruff keys at a local hardware store to replace the one if it is missing or broken.
If is not a broken woodruff key then look at the larger pulley wheel the belt is on and see if there is a twist type lever or a slider type lever that makes the moped "free wheeling" like a bicycle. This theory is a maybe because you mentioned chains on both sides so the free wheeling unit is usually on the rear hub like a bicycle.