The control board appears to use brown + blue for supply, and the red wire likely looks like the switched output/live to the blue socket.
So, to answer your question:
No, PE should not be treated like L or N on the control board side.
Normally for a unit like this:
• L (live) goes to the controller input
• N (neutral) goes to the controller input
• PE (earth) does not go to the PCB as a functional conductor unless the board specifically has an earth terminal
• PE should be joined through from supply cable to the output/socket earth terminal
Based on your pictures, the terminal block seems to be acting as a junction point:
• one side for incoming mains
• one side for the socket/output
• then the PCB takes L/N, and returns a switched live on the red wire
That part is plausible.
What is not okay is the current state of the exposed conductors and uncertain earth routing.
About the missing temperature sensor:
• Yes, if this is a GSE fan controller that depends on an external temp sensor, then without the sensor it may not regulate correctly
• Depending on the model, it may:
• do nothing,
• run the fan at minimum,
• run full speed,
• or behave erratically
So the missing sensor could absolutely make it seem “without function,” but it does not explain unsafe-looking wiring.
My honest take from the photos:
• It may not be defective
• but it is not safely wired right now
• and without the sensor, it may also be incomplete / impossible to test properly
If this still doesn’t help you out I would highly recommend posting on a local electricians page on Facebook. That would be your best bet and probably have your question answered quickly. Best of luck!