VW CC 2.0t 2010 - Car stutters and turns off when engine reaches normal temp

Hello,

I have a VW CC 2.0t 2010 that i bought a few months ago and it ran fine for about 3-4 months.
The first issue happened when cold weather started.
It was a battery problem, my car would not start if i let it sit for the night.
I bought a new battery and the issue was fixed.

The car ran fine again for approximately 1 month, with no issue.

Then, suddenly a strange problem started occurring. I would drive for over 10 minutes, and for example stop to go get gas, turn off my car, gas up, and when I tried turning the car back on, it wouldn't, it just stuttered, started for a second and shut off.
I had to wait 1 hour before my car would start again.
A similar problem occurred but this time, I did not turn off the car, I left it on park and after a few seconds or a minutes, it stuttered and shut off, and then same issue, have to wait 1 hour before I can start it again.

So I wanted to diagnose this issue, I noticed that the car drives fine and doesn't turn off, if the engine is cold, this issue seems to be happening only when the engine's temperature reaches ~90 or a little less but close to that normal engine temp, which is where it should be normally.
I also tried to pump the gas pedal when I was feeling the stuttering, and it seems like sometimes it could keep the engine going for a few more seconds, but it would still just shut off eventually.

I had the check engine lit when all this happened.

I brought the car to the garage for a diagnostic.
The mechanic said that he read some codes from the car, he said it was related to the low pressure fuel pump, high pressure fuel pump and control module.

My dad is a handyman and we could get our hands on these parts, so we wanted to do it on our own.
So I bought the control module, replaced it, same issue.
I bought the low pressure fuel pump, replaced it, same issue.
I then bought the high pressure fuel pump, replaced it and the same issue is still happening, when it reaches the normal temp, stutters and shuts off.

I'm running out of ideas.

Is there anyway you can help me solve this problem please?