Modern Vespa

So I recently purchased a Piaggio Ciao 1974 model with turn signals (in great condition with <900 miles). This is my first mechanical project but I'm a product designer so I didn't think it would be too much to manage. When I got it, it would start up no problem but with a major caveat, in that it would not stay running when I came to a full stop or sometimes when I first took it off the stand. As a result, here's what I did:

Cleaned the carb

Replaced the spark plug

Cleaned the air filter

Replaced the petcock

Replaced the battery

Filled with fuel + oil mix

Adjusted throttle screw to be higher (now it will at least idle with the lights off)

P.s. yes the brake light does function because it turns on just fine.

So, thinking I'd fixed the problem, I took her for a spin. The moped started, rode great but with no lights. Lights will eventually kill the engine especially at stops. Mostly at stops. So I rode it without the lights turned on and got it running pretty well. When I stopped the moped wreaked of a burning smell, the exhaust was burning hot and when I went to start it again.. nothing. It took some serious waiting it out and trying repeatedly to finally get it started. Since then, its gotten even harder to start and turning on the lights is a sure fire way to make your next stop longer than longer than anticipated.

I've read quite alot of suggestions especially regarding the cleaning of the carb. I have recently purchased an altogether new 12/10 Dellorto carb to replace mine, just in case but I don't see why this would have to do with the electrical. Some have suggested checking the breaker points while other have suggested replacing the stator and/or condenser.

I'm a bit anxious to open the engine first the first time so I'd like to try all of my options before doing so. Can anyone point me in the right direction as to what to inspect next?

Is it a single speed Ciao or a variated model? If it has side covers, the single speeds only have one circular protrusion on the left side cover and the variated ones have two circular protrusions, one in front and one in the back.

Pretty sure the lights run off the coils and the battery is there solely for the turn signals.

There is no stator plate. The coils and points bolt directly to the crankcase and are fixed in position. You advance/retard timing via the point gap. Hope you don't need to change anything, because you'll have to pull the front clutch (single speed) or variator off, the exhaust and drop the motor out of the chassis to remove the fan and expose the fan and coils beneath it. Not a hard job, but a bit frustrating if you're new to Vespa mopeds.

If it's single speed, I would suspect that the bearings in the clutch bell are dried out and the clutch isn't freewheeling on the crank. It's also possible your centrifugal clutch isn't engaging and releasing properly.

Another area to check is the decompressor valve on the head. It's the cable that goes over near the spark plug. That can crud up and stick, causing compression loss at idle and lean conditions (hot motor) at speed. You may want to clean it with some carb cleaner and actuate the lever (start lever) a bit to see if it moves freely.

I'll try to get some pictures in the daylight, I have an '81 Ciao SC in the garage.

Alot of people have mentioned the compression and cleaning this area. Will I need to remove the entire front piece of the engine (the cyclinder?) in order to clean the piece you are referring to. Also I have the single speed model with only one circle on the side coverings. Thanks so much for your reply! This has been immensely helpful.