Modern Vespa

Took the engine out, rechecked the valves, rechecked compression - 150

decompression spring is in place. Timing looks good.

Does everything look ok here?

Decomp mechanism is centrifugal so should close as soon as engine starts, could picture a scenario where it's sticking at low rpm though, could trouble shoot by removing it, 150s don't really need it. decompression mechanism removal

My money's still on the carb though, sold mine or would donate it to the cause.

Backfiring to me seems like a timing issue since you seem to have been very systematic about ruling out the other usual suspects.

But if the engine ran poorly but stopped running altogether after a jet change…I too would continue to be suspicious of the carb before I started messing around with the decompression spring. What happens if you put the old jet back in?

Re your pics. An issue I had when mine was stock was belt slippage. Symptom was hesitation off the line, not stalling just hesitation in acceleration. When I took the cover off it was full of belt dust. Fixed with a stiffer contra spring. Yours looks fine.

Gents, thank you so much for keeping an eye on this thread and helping - I appreciate your expertise and advice.

Out of hopelessness and frustration - went and bought a new spark plug. Put it in - voila - the scooter fired right up. I believe, when I went and bought the previous plug, the catalogue at the shop showed a different plug (I had no idea this would make such a huge difference).
Now, I am back to rough idle issue.
Took it on a few mile ride, it was idling odd - sounds like a Harley🤣, but, no issues accelerating.
Will to wiki to see how to adjust idle screw and the mixture screw, as I am now a 100% sure my process is not right.
Here's a video of it idling. It sounds to me that the idle screw is turned waaay in and when I change the fuel/air screw ratio - nothing really changes.

https://m.youtube.com/shorts/qz6OujIR3xQ

sounds quite good actually!

To set the mixture properly - get the correct carb screwdriver first! Then turn it all the way in, then turn it 2.5 turns out. Then set your idle screw to an idle that sounds good. Then go back to the mixture screw and turn it out more and see if the idle speed increases. Likely it doesn't. Turn mixture screw in until idle speed decreases. Essentially you are tuning the mixture to get the highest idle speed at a given idle screw setting. Once you have the mixture screw at that optimal point, then you can back off the idle screw to a speed which not rough, but not fast. BTW, before you try any of this, you need to warm the engine up a lot - drive it for 15 minutes or more. Setting the mixture and idle on a cold engine will make for idle problems when the engine is warm.

About your spark plug issue: I bought some idle jets a while back, the stores that sold them to me where vespa specialty stores. When I installed them, my lx50 wouldn't even start. In one case I had to turn the mixture to full in, and the idle screw to max in, just to get it to start. What was the problem? The new idle jets didn't screw into the cavity all the way, and so were basically not putting much fuel into the system. So, lesson is that you need to do your homework about parts, not rely on any partsman at a store.

Pulled the carb, ultrasonic cleaning, back together - scoot starts every time.
Once I get above 8-10 mph - does great, however, the acceleration lag persists. Rode it for about 20 minutes, tried every possible setting of the air/fuel mixture screw to no avail. Got it to idle perfectly - yet, when I accelerate from a stop - it lags.