Modern Vespa

My GTS300 from 2011 was not starting when I bought it spring last year.
Old owner had also bought it like this and had tried lot of things to get it start.
- Change fuel pump (new mounted)
- Fuel injector (swapped from other scooter)
- Throttle control (bought from eBay)
When I got the scooter I found the fault, the register chain was 2 teeth wrong.
So after mount it right it start. I have some issues at startup when it's "cold". I wonder if the last owners tried to adjust "idle screw" (TPS) since it not going like it should. I got the Throttle control with when I bought the scooter and I wonder to change to that one since the fault was something else when it last time was mounted.
So the Throttle control I have is including some other parts I think I also need to mount.
Please help me out regarding the parts I have, it is what I need to get it start with the other Throttle control?
Do I need other parts, or do I not need to change them all?
(more pictures can be added if needed)

GTS300s are fuel injected, so you won't need the top item in your photo, that's a carburetor.

looks like a throttle body to me. I don't see a carb.

You are probably correct. Thanks for the clarification.

That screw on the throttle body / ecu should never ever be touched.

Whichever throttle body you install, you must use the key with a matching 'chip' in it. So if you swap then you'll need to use a different key.

Bad idle can be caused by a dirty idle bypass inside the throttle body (spray throttle body cleaner in it). Can also be caused by air leaks where the throttle body is connected to both black rubber hoses (check correctly seated and clamps tight). A bad air filter can also cause problems as well as a bad seal in the join between the two halves of the airbox.