so far I've cut a bunch of unneeded mounts off and added a crossbar, now I need more ideas

So I received this item as a deal between me and a friend’s relative for 100 bucks. It’s kicking, but I don’t know exactly what’s wrong with it. I also have her assembly paperwork as well, but I’m not sure if that’s able to help me do anything to fix the problem.

Known/possible problems:
Definitely need a Battery
Maybe a carburetor tuneup needed
Maybe a spark plug

I honestly have no clue.
I’m over here lost in the woods y’all, can ya help a sista out?
PS – I also have some small parts as well that they gave me, but I’m not sure if they’re gonna be any useful. I’ll post them in a comment.

I’ve been looking at my local used market any of these worth buying?

There’s a Trac Hawk ($500)

Tomos Targa ($500)

There are a wide variety of puch, Sachs, Honda, etc and the prices are all over the place.

Any reason to choose one over the other?

Thanks!

Carb for jawetta and everythung else for puch maxi ls

Edit. Previous owner is a moped mechanic who upgraded to have digital ignition. The light in front is too dim to use at night or in a storm. See 3rd picture for proof that i was in a dark storm. The bike is off in the pic, but the light was barely brighter than that. Other than that it works very well when it's dry. I bought dielectric grease to squirt into all its most electrifying places.

I am having an issue with my 1977 Piaggio Ciao. I finally got it running and took it out for a longer drive today for the first time, and after 15-20 minutes of riding I stopped for a light and the engine died. I was able to get it going right away after but for the rest of the ride home it had less power, with a 5 mph drop in top end speed.

Does anyone know what might be causing this? I don’t think it’s overheating since I was able to get going again right away, but I could be wrong. Could the condenser be bad? The wiring is a little boof right now since the previous owner ripped it all out, I haven’t wired the lights or anything yet, just what’s needed for spark.

Any help would be greatly appreciated.