Modern Vespa

A little background
I bought a police repo 2009 range 2t Vespa lx 50
This is a Euro version with no kill switches and Km/H on the speedometer
Bought in bad shape. No ignition, scratches on the side where I assume it was dropped. No spark or any power for that matter.
The scooter runs now and starts on the first kick.
I have replaced the following:
Relay, battery, fuse,ignition, Voltage regulator, CDI, Stator with woodruff key, spark plug and cable, carburetor, fuel lines, vacuum lines, oil lines, fuel petcock, more wires are new than not at this point, and throttle cable. As well as lots of cosmetic effects
Today the scooter starts up and runs best with the mix valve closed, it will still dies on full throttle but drives pretty fast before it shuts off at full throttle.
With the mix valve open 1.5 turns it dies on throttle , runs well cold and dies on hills or after a few minutes driving.
I have it set at 1.5 turns from closed on the video
When the scooter warms up, there is lots of blue smoke, filling the driveway and pouring out all around the exhaust and out from the end.
Am I wrong in thinking this may be solved by disabling the oil auto mix and mixing the fuel myself?

When you replaced the throttle cables and hoses, did you reset the oil pump? There's two marks (one on the case, one on the pump) that need to line up. You can inspect them via the black rubber plug below the belt cover.

MjRally, This is a great idea! I changed the throttle cable from the splitter to the carburetor only. I did not realize that was a thing to look at.
I looked and I don't see markings. I'm sure I'm missing something here

Wipe the area all down. Here's a photo with one mark clearly visible. It's not the best photo, but look in the viscinity.

MJRally, you were correct the lines were there and not lined up. I attempted to adjust the line and found the adjustment screw is snapped in two.

That's a great catch. Thanks for the tip!
I removed it and will report the fix tomorrow if the part is available at an autopart store or scooter shop nearby

Smoke problem fixed! I was able to repair the screw for the time being. It is about 97% fixed. Almost no more smoke at all.

The scooter still dies on full throttle. And dies going up hill as well. Specifically it slows until it shuts down going up hill. Down hill it runs great and fast needs very little gas going down hill.
Any idea what could be causing that?
In the manual it lists a few Carb issues. I've gone over the carb a few times, I think it's running well.
I am going to check the SAS filter now and tank breather hole.
I'm also going to have to bleed out the air in the line. I changed the oil and I can see the new full Syn purple oil is moving along but there are big air bubbles. I added a cap of 2T to the gas to make sure there isn't a problem while the air passes.
Could it be the carb flooding? I bought the carb on eBay from Valcotrade Carburettor to fit Vespa LX50 LXV 50cc S, 17.5mm, 2 Stroke 50, 2005 - 2013 Carb. It was near perfect to match to what was on the scooter.

Any other ideas?

Possibly leaning out at full throttle? Check the float height, it may be set low causing lean running at full throttle.

Just a guess, I haven't had a bike with a carb in years.

I gave it a try. Nothing much happened. Still no power up hill.
The scooter runs best with the mixture screw almost all the way in.
The scooter is pretty loud, I guess someone put an exhaust kit on it. But there is a concerning snapping, sounds like plastic hitting not knocking I think.
Jumpshare - Noise on LX 50.mp4

Let's see a picture of the exhaust side.

As far as the oil pump air bubbles. Bleed those out asap before you have problems! They wont push through the system like other oil pumps.

As far as the carb goes, did you check the jets and needle inside the eBay carb? Just cause the outside looks close doesnt mean the internals are.

An aftermarket pipe often means lean running unless the carb is jetted to match. It'll also mean hotter running.

Thanks so much for the response. Attached is a picture of the exhaust side
I'm going to bleed out the bubbles today.

I have been suspicious of the carb since I got it. The scooter runs best when it is cold and running off the automatic choke. When I was attaching the slide I noticed it was a different size to the one that was on the scooter when I got the scooter. It was the original DELLORTO PHVA on the scooter when I got it.
Do you think I should return the carburetor and buy the original carburetor DELL'ORTO PHVA 17,5? I guess I'm wondering if that could do harm. I have been thinking about doing this. It is an $80 difference and well within my budget.
The tricky thing is I don't know if they bored out the engine or did other things to it.

Running hot and loud is what made me think it was either an aftermarket exhaust or altered in some way to maybe add power.

Even when running the best I have got it to run it does not seem more powerful than expected though.

No photos attached

Here is the exhaust photo
Not sure why it failed before

Exhaust looks like the Piaggio stock. I doubt your motor is bored out or has had any modifications to it. Have you crawled under it and inspected the manifold for leaks? That would make a louder sounding exhaust/ drip oil/ poor performance. Make sure everything is secure!

Do you still have the original carb? Open it up and compare apples to apples with the EBay carb. Dont swap anything just yet. Just get some data.

Going back to your plastic snapping noises. There is a plastic fan on the exhaust side. Do you see any debris in the grill?

Regarding the manifold, I crawled under the scooter and inspected the manifold for cracks. I did not see anything. I also sprayed soapy water and did not detect any leaks.
On the carburetor, I do have the original carburetor. The jets are significantly smaller and the slide is smaller on the original. The throttle cable that came with the eBay carb did not reach the splitter, I used the original cable to make it fit. On retrospect, that should have been an indicator to just move on from the Ebay carb.
With regard to the exhaust fan. There does not seem to be any debris or hitting the fan. The noise is hard to predict and replicate but I think this is where the noise is coming from. I'll check carefully next time it happens.

I bled out the air bubbles. I attached photos in case someone else is bleeding air bubbles. In the service manual it shows the screw but does not give advice on how to catch the dripping oil. The oil drips behind the cable and would run into the back of the crankcase if it was not caught. I used 2 water bottles connected with one cut to fit under the dripping screw. Kept the crankcase free from oil. You can see on the first photo the marked lines on the oil pump are pretty well lined up now. I'm still waiting on the new adjustment screw to line them up perfectly.

I changed the carb back to the original type.
The only trouble I am having now is the scooter runs strong when cold and takes a full throttle.

Not much changed. I checked again for vacuum leaks both with soapy water and starter fluid. There does not seem to be a vacuum leak. I also checked the fuel line when I changed it to a clear line to be sure the throttle sends the fuel into the line. I can confirm I am getting fuel and oil into the lines. The oil is full syn 2 stroke oil. I chose a purple color so I can see it going into the carburetor. I did a full oil and gas change as well as the petcock for the fuel line.

The scooter starts right up and idles well.
I can drive right out of my driveway running perfectly but it will not make it around the block before it stops taking throttle. After a couple minutes it only takes half throttle and dies on a small incline.
After it dies the scooter starts right up but has little power on flat ground, no power to go up a hill and only take maximum 50% throttle in any conditions.

I attached a video link that shows how it idles and what it does on throttle.

https://jmp.sh/iXhbwVYl