Okay first off:
The reason you can't connect just one wire from stator to rectifier is because AC current goes back and fourth. Technically it goes up and down with the sine wave. If you connect one wire, you will only get half the voltage rectified. Because half the sine wave will enter the rectifier, but the other half will go towards the frame, since the other wire is grounded there. This stuff is confusing. Unless someone on here can recommend how to use frame ground, I would stick to what I'm showing you below.
Notice the two circuits circled
You can see that you have created an independent DC circuit for all the stuff you want to power! The rest stays AC to the frame.
These drawings are simplified. But I hope the big picture makes sense.
Here is how your wiring should look like now that we've understood why this person has wired their moped this way.
Let me know if I can improve this post. I will now be working on my own wiring diagram but it will prob look like the one above.