I'm gonna be honest with you. I don't see any possible way for a dead Nvidia GPU on Optimus to be messing with your screen.
The HD4600 handles everything from the moment you press the power on button, and I doubt really hard that it is dead. The nvidia GPU is connected to the Intel GPU, regardless of it's state the computer should boot up atleast.
All you have to do is disable the Nvidia in the device manager and it would use everything on the Intel.
Are you sure you don't have a broken wire for the lcd screen, short circuit etc?
Are you sure that specific shop you spoke with have electronic capability to evaluate your computer?
I'm not into electronics, but from my work, studies and experience the Intel GPU always turns on. Thats how Optimus works, the GPU image comes always from the HD4600, which is rerouting the Nvidia.Trusted Reviews:How Optimus Works
Not saying they aren't correct, but there is more to it than just the Nvidia card not working.